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Decomposition of Graphs into PathsFabio Botler and Yoshiko Wakabayashi

E-mail: {fbotler,yw}@ime.usp.br

Abstract—A decomposition of a graph G is a set D = {H1, · · · , Hk} of pairwise edge-disjoint subgraphs of G that cover the set ofedges of G. If Hi is isomorphic to a fixed graph H, for 1 ≤ i ≤ k, then we say that D is an H-decomposition of G. In this work, westudy the case where H is a path of fixed length. For that, we first decompose the given graph into trails, and then we use adisentangling lemma, that allows us to transform this decomposition into one consisting only of paths. With this approach, we tacklethree conjectures on H-decomposition of graphs and obtain results for the case H = P` is the path of length `. Two of these resultssolve weakenings of a conjecture of Kouider and Lonc (1999) and a conjecture of Favaron, Genest and Kouider (2010), both for regulargraphs. We prove that, for every positive integer `, (i) there is a positive integer m0 such that, if G is a 2m`-regular graph with m ≥ m0,then G admits a P`-decomposition; (ii) if ` is odd, there is a positive integer m0 such that, if G is an m`-regular graph with m ≥ m0 andcontaining an m-factor, then G admits a P`-decomposition. The third result concerns highly edge-connected graphs: there is a positiveinteger k` such that if G is a k`-edge-connected graph whose number of edges is divisible by `, then G admits a P`-decomposition.This result verifies for paths the Decomposition Conjecture of Barat and Thomassen (2006), on trees. This work is an extendedabstract of the Ph.D. thesis of the first author, written under the supervision of the second author.

Index Terms—graph, path decomposition, highly edge-connected, regular graph.

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1 INTRODUCTION

A decomposition D = {H1, . . . ,Hk} of a graph G is a set ofedge-disjoint subgraphs of G that cover the edge set of G.An implication of Hall’s Theorem (1935) states that bipartiteregular graphs admit a decomposition into perfect match-ings. A consequence of this fact, already known by Petersenin 1891, is that even regular graphs admit decompositionsinto 2-factors. Here, we say that a graph G is even (resp. odd)if every vertex of G has even degree (resp. odd degree).

Since then, many results on decompositions have ap-peared in the literature. For example, Pyber [41] proved thatevery simple graph can be decomposed into at most fourodd graphs. Junger, Reinelt e Pulleyblank [30] considereddecompositions into connected subgraphs with k edges(and at most one subgraph with less than k edges), calledk-partitions. They proved that every k-edge-connected graphadmits a (k+ 1)-partition, for k = 1, 2, 3, and every 4-edge-connected graph admits an s-partition, for every positiveinteger s. They also conjectured that 2-edge-connected pla-nar graphs admit 3-partitions composed only by trails.

In this work we are interested in decompositions D ={H1, . . . ,Hk} of a graph G in which Hi is isomorphic to afixed graph H , for every 1 ≤ i ≤ k. Such a decompositionis called an H-decomposition. As noted by Haggkvist [25], anatural question is, given two graphs G and H , to decideif G admits an H-decomposition.

“Given two graphs G and H and an inquis-itive mind we may ask whether or not Gis the edge-disjoint union of copies of H .”(HAGGKVIST, 1989)

This research has been partially supported by CNPq Projects (Proc.477203/2012-4 and 456792/2014-7), Fapesp Project (Proc. 2013/03447-6) andMaCLinC Project of Numec/USP, Brazil. F. Botler is supported by Fapesp(Proc. 2014/01460-8 and 2011/08033-0), and Y. Wakabayashi is partiallysupported by CNPq Grant (Proc. 303987/2010-3)

An obvious necessary condition is that |E(G)| is divisibleby |E(H)|. Junger, Reinelt and Pulleyblank’s result aboveimplies that if G is connected and has an even number ofedges, then G admits a P2-decomposition, where P2 is thepath with two edges. On the other hand, Dor and Tarsi [17]proved that when H is connected and has at least 3 edges,the problem of deciding if G admits an H-decomposition isNP-complete. Thus, it is natural to search for sufficient con-ditions to obtain such decompositions. For decompositionsinto triangles, for example, Nash-Williams [38] conjecturedthat minimum degree 3n/4 is sufficient, i.e., if G is a graphon n vertices and minimum degree δ(G) ≥ 3n/4 (and|E(G)| is divisible by 3), thenG admits aK3-decomposition.To approach Nash-Williams’ Conjecture, Barber, Kuhn, Loand Osthus [4] proved that even graphs with minimumdegree δ(G) ≥ 9n/10 + o(n) and such that |E(G)| isdivisible by 3 admit K3-decompositions.

In the case where H = T is a tree, Barat andThomassen [3] conjectured that high edge-connectivity issufficient, i.e., they proposed the following conjecture.

Conjecture 1.1 (Barat–Thomassen, 2006). For each tree T ,there is a positive integer kT such that, if G is a kT -edge-connected graph and |E(G)| is divisible by |E(T )|, thenG admitsa T -decomposition.

Barat and Thomassen’s Conjecture, known as the De-composition Conjecture, is the main subject of this work.Thomassen [46], [47], [48], [49], [50] proved this conjecturefor stars, some bistars, paths of length 3, and paths oflength 2k, for every positive integer k. In this work, weprove that this conjecture holds for paths of every fixedlength.

To prove Barat and Thomassen’s Conjecture (for paths),we first study T -decompositions of regular graphs. In 1964,Ringel [42] conjectured that the complete graph K2`+1

admits a T -decomposition for any tree T with ` edges.

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Ringel’s Conjecture is commonly confused with the GracefulTree Conjecture that says that every tree T with n verticesadmits a labeling f : V (T ) → {0, . . . , n − 1} such that{1, . . . , n − 1} ⊆ {|f(x) − f(y)| : xy ∈ E(T )}. since theGraceful Tree Conjecture implies Ringel’s Conjecture [43],Ringel’s Conjecture holds for many classes of trees, such asstars, paths, bistars, carterpillars, and lobsters (see [18], [24]).Haggkvist [25] generalized Ringel’s Conjecture for regulargraphs.

Conjecture 1.2 (Graham–Haggkvist, 1989). For each tree Twith ` edges, if G is a 2`-regular graph, then G admits a T -decomposition.

Haggkvist [25] also proved that Conjecture 1.2 holdswhen the girth of G is at least the diameter of T . Formore results on decompositions of regular graphs into trees,see [20], [22], [27], [29], [44]. In the particular case whereH = P` is the path with ` edges and G is a regular graph,Kouider and Lonc [32] improved Haggkvist’s result, prov-ing that a 2`-regular graphGwith girth g ≥ (`+3)/2 admitsa P`-decompositionD such that every vertex ofG is the end-vertex of exactly two paths of D. They also conjectured thatthis fact must hold for every 2`-regular graph, i.e., that every2`-regular graph G admits a P`-decomposition D such thateach vertex of G is the end-vertex of exactly two paths of D.In Section 4, we prove a weakening of Kouider and Lonc’sConjecture. We prove that, for each positive integers ` and gsuch that g ≥ 3, there is m0 = m0(`, g) such that, if G is a2m`-regular graph with m ≥ m0 and girth at least g, then Gadmits a P`-decomposition D such that each vertex of G isthe end-vertex of exactly 2m paths of D.

Another result related to the result stated above is due toKotzig [31], that proved that a 3-regular graph G admits aP3-decomposition if and only if G contains a perfect match-ing. Favaron, Genest, and Kouider [21] conjectured that thisresult may be generalized, i.e., that odd `-regular graphsthat contain perfect matchings admit P`-decompositions. Inthis case, the degree of the vertices of the graph is decreasedby one-half, but a perfect matching is required. In Section 4,we prove a weakening of Favaron, Genest, e Kouider’sConjecture. We prove that, for each positive integers ` and gsuch that ` is odd and g ≥ 3, there is m0 = m0(`, g) suchthat, if G is an m`-regular graph with m ≥ m0, girth atleast g, and containing an m-factor, then G admits a P`-decomposition D such that every vertex of G is the end-vertex of exactly m paths of D. Finally, as suggested byHaggkvist, we conjecture that this fact may hold in a moregeneral way, with m0 = 1.

“It is a trend in modern mathematics that if youcan not solve a particular problem you can at leastformulate a more general one, thus increasing theamount of frustration in the world and keepingyour collegues on their toes.” (HAGGKVIST, 1989)

The main tool developed in this work is a disentanglinglemma (see Section 3), that showed to be useful to dealwith Graham and Haggkvist’s, and Favaron, Genest, andKouider’s Conjectures in Section 4, as well as Barat andThomassen’s Conjecture in Section 5. Another importantconcept is the one of complete decomposition, first introducedin [12], and that will be used frequently in this work.

Roughly speaking, here the completeness (of a decomposi-tion) represents a special set of properties of a decomposi-tion. Throughout our proofs, we develop an algorithm thatreceives a complete decomposition D and returns a “better”complete decomposition D′ (in the sense that D′ is closerto the desired decomposition). Therefore, given a graph Gwe first construct a decomposition D of G, prove that D isa complete decomposition, and use the algorithm to obtainthe desired decomposition.

In Section 2, we present briefly the concepts aboutgraphs that are more important in this work. In the fol-lowing sections, we reorganize the results in [10], [11] asfollows. In Section 3, we present a disentangling lemma(Lemma 3.2), that allows us, under some conditions, toswitch edges between elements of a decomposition of agraph G into trails and obtain a new decomposition of Ginto paths. In Section 4, we use Lemma 3.2 to obtain decom-positions into paths of fixed length of a large family of regu-lar graphs. In Section 5, we generalize part of the techniquedeveloped by Thomassen that deals with decompositions ofhighly edge-connected graphs. This technique give us thestructure needed to apply some of the lemmas obtained inSection 4, and prove Barat and Thomassen’s Conjecture forpaths of any fixed length.

2 NOTATION AND BASIC CONCEPTS

The basic terminology and the notation used in this workare standard [7], [16]. A multigraph is a pair G = (V,E). Thesets V and E, also denoted by V (G) and E(G), are, respec-tively, the sets of vertices and of edges of G. We say that amultigraph is simple or, simply, a graph if it does not containparallel edges. All graphs and multigraphs considered hereare finite and have no loops. In this work we use mainlygraphs, except in Section 5, more specifically, in the proof ofLemma 5.17. Thus, the following definitions are presentedfor graphs, but can easily be extended for multigraphs.

We denote by dG(v) the degree of a vertex v ∈ Vand, when G is clear from context, we write only d(v).If d(v) = 0, then we say that v is an isolated vertex in G.If d(v) = r for every vertex v of G, then we say that Gis r-regular. We say that a graph H is a subgraph of G ifV (H) ⊆ V (G) and E(H) ⊆ E(G), and we use frequentlythe arithmetic operators + and − in the following way. If Fis a set of edges, and e is an edge, we denote by F + eand F − e the sets F ∪ {e} and F \ {e}, respectively. If H isa subgraph of G, and e is an edge of G, then we denote byH+e the graph obtained from (V (H), E(H)+e) by addingthe vertices of e that are not present in V (H); and by H − ethe graph obtained from (V (H), E(H)−e) by removing thevertices incident to e that become isolated. Given F ⊆ E,we denote by G[F ] the subgraph of G induced by F , i.e., thegraph obtained from

(V (G), F

)by removing the isolated

vertices. We also denote by dF (v) the number of edges of Fincident to v or, equivalently, the degree of v in G[F ].

Let G = (V,E) be a graph. A trail in G is a subgraphof G that admits a sequence of vertices T = v0v1 · · · v` suchthat vivi+1 ∈ E, for 0 ≤ i ≤ ` − 1, and vivi+1 6= vjvj+1,for 0 ≤ i < j ≤ ` − 1. It is also convenient to refer to atrail T = v0v1 · · · v` as the subgraph of G induced by theedges vivi+1 for i = 0, . . . , ` − 1. The length of a trail is

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its number of edges. We also say that a trail of length `is an `-trail. A path in G is a trail that admits a sequenceP = v0v1 · · · v` such that vi 6= vj , for 0 ≤ i < j ≤ `.We also say that v0 and v` are the end-vertices of P , andthat P joins v0 to v`. A cycle is a trail that admits a sequenceC = v0v1 · · · v`−1v0 such that vi 6= vj , for 0 ≤ i < j ≤ `− 1.The path of length `, also called the `-path, is denoted by P`(this notation is not standard). Given two vertices, u and vin G, the distance between u and v is the length of a shortestpath joining u and v in G; the diameter of G is the largestdistance between two vertices of G; and the girth of G is thelength of a shortest cycle in G.

An orientation O of a subset F ⊆ E, is an assignmentof a direction (from a vertex to the other) to each edgeof F . If an edge e = uv in F is directed from u to v, wesay that e leaves u and enters v. Given a vertex v of G, wedenote by d+

O(v) (resp. d−O(v)) the number of edges of F thatleave (resp. enter) v with respect to O. In this work we usethe term Eulerian graph as a synonym of even graph, i.e., anEulerian graph is a graph that contains only vertices of evendegree. In other words, here we allow an Eulerian graph tobe disconnected, and avoid using the term even to refer tographs. An Eulerian orientation of an Eulerian graph G is anorientation O of E(G) such that d+

O(v) = d−O(v) for everyvertex v in V . Moreover, we say that a subset F ⊆ E isEulerian if G[F ] is Eulerian. We denote by G = (A,B;E) abipartite graph G with bipartition A,B of its set of vertices.

We say that a set {H1, . . . ,Hk} of subgraphs of G is a de-composition of G if

⋃ki=1E(Hi) = E and E(Hi)∩E(Hj) = ∅

for every 1 ≤ i < j ≤ k. LetH be a family of graphs. AnH-decomposition D of G is a decomposition of G such that eachelement of D is isomorphic to an element of H. Moreover, ifH = {H}, then we say that D is an H-decomposition.

We say that a subgraph H of G is spanning or a factorof G if V (H) = V (G). A factorization of a graph G is adecomposition of G into factors. A k-factor of G is a k-regular factor of G, and a k-factorization of G is a decom-position of G into k-factors. The following result, Petersen’s2-Factorization Theorem [40], is essential for the inductionsteps in our proofs.

Theorem 2.1 (Petersen, 1891). If G is a 2`-regular graph,then G admits a 2-factorization.

Let G be a connected graph. The edge-connectivity of G isthe smallest integer k such that there is a set F ⊆ E(G) with|F | = k and such that G−F is disconnected. If G has edge-connectivity k, then we say that G is k-edge-connected. Thecelebrated Menger’s Theorem [35] says that given a positiveinteger k, a graph G is k-edge-connected if and only if forevery pair of vertices u and v there are at least k pairwiseedge-disjoint paths joining u to v in G.

The remaining concepts that we need will be presentedin the forthcoming sections. In Section 3 we define tracking,used to specify the order in which the vertices of a trailis visited. In Section 5 we define fractional factorizations andbifactorizations that extend the concept of factorization, intro-duced to deal with highly edge-connected bipartite graphs.

3 A DISENTANGLING LEMMA

To decompose a graph G into copies of a fixed graph H , weuse the following strategy: we obtain first a decomposition

of G into elements similar to H ; afterwards, if needed,we perform edge-switchings and obtain copies of H . Forexample, to decompose a graph into paths of length `, wecan find first a decomposition D into trails of length `. Ifthe trails in D have no cycles, then D is a decompositioninto paths. This occurs, for example, in the case of graphswith large girth, more specifically, girth larger than `. If inD there are trails that are not paths, we use disentanglingtechniques, this is, an edge-switching step (between theelements of D), in order to find a new decomposition D′with less elements that contain cycles.

Heinrich, Liu, e Yu [26] show how to decompose 3m-regular graphs that contain an m-factor into paths oflength 3. For that, they construct a decomposition of thegraph into trails with 3 edges, i.e., 3-paths and triangles, andshow that if in this decomposition there is any triangle X ,then there is another element Y such that X ∪ Y can be de-composed into two paths of length 3. In [12], we decomposea 5-regular graph, that contain a perfect matching and hasno triangles, into trails of length 5 that may contain a C4 (seeFigure 1). Then we show that if we switch edges between theelements of this decomposition in a specific order, we obtaina decomposition into 5-paths. In [32], Kouider and Loncdecompose 2`-regular graphs with girth at least (` + 3)/2into trails of length `, and show that for each trail X0 thatis not a path there is a sequence X0X1 · · ·Xk of elementsof this decomposition such that it is possible two switch anedge of Xi with Xi+1 (i = 0, . . . , k − 1) and obtain onlypaths of length `.

Fig. 1: Trails of length 5 in triangle-free graphs.

In this section we present a disentangling lemma thatwe developed [10] for paths of any fixed length. Morespecifically, we prove a result, Lemma 3.2, that guaranteesthat, given a decomposition of a graph into certain specialtrails, it is possible to switch edges between the elementsof this decomposition and construct a decomposition withmore paths (that preserves some properties). Our techniqueis similar to the technique introduced by Kouider andLonc [32], however here we make use of a sufficiently highminimum degree to compensate the requirement of a largegirth.

3.1 Trails, trackings, and augmenting sequencesWe recall that a trail is a graph T for which there is asequence B = x0 · · ·x` of its vertices (possibly with repe-titions) such that E(T ) = {xixi+1 : 0 ≤ i ≤ ` − 1} andxixi+1 6= xjxj+1, for i 6= j. Such a sequence B of vertices iscalled a tracking of T , and we say that T is the trail inducedby the tracking B. We say that the vertices x0 and x` are theend-vertices of B. Note that a path admits only two possibletrackings, while a cycle of length ` admits 2` trackings.

Given a tracking B = x0 · · ·x`, we denote by B− thetracking x` · · ·x0, and, for ease of notation, we denoteby V (B) and E(B) the sets {x0, . . . , x`} of vertices and{xixi+1 : 0 ≤ i ≤ ` − 1} of edges of B, respectively.

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Moreover, we denote by B the trail(V (B), E(B)

). It is

convenient to say that a tracking B = x0 · · ·x` traverses thevertices x0, . . . , x` and the edges x0x1, . . . , x`−1x` (in theseorders), and that x0x1 is the initial edge of B and x`−1x`is the final edge of B, or that B starts with x0x1 andends with x`−1x`. A set B of edge-disjoint trackings of agraph G is a tracking decomposition (or decomposition intotrackings) of G if

⋃B∈B E(B) = E(G). Note that the set

B = {B : B ∈ B}, called decomposition subjacent to B, isa decomposition of G into trails. We also note that if Biand Bj are trackings of a tracking decomposition B suchthat E(Bi) ∩ E(Bj) 6= ∅, then Bi = Bj and, consequently,Bi = Bj (i.e., Bi and Bj induce the same trail).

We say that a tracking x0x1 · · ·x` is a vanilla tracking ifthe tracking x1 · · ·x`−1 induces a path in G, and we saythat a trail T is a vanilla trail if there is a vanilla trackingof T (see Figure 2). If a vanilla tracking (resp. vanilla trail)contains ` edges, then we say that it is an `-vanilla tracking(resp. `-vanilla trail). Here we fix the central object of thiswork: we say that a tracking decomposition B is an `-decomposition if every element of B is an `-vanilla tracking.Note that we do not carry the terms “tracking” or “vanilla”in `-decomposition, i.e., in this text, every element of an `-decomposition is a vanilla tracking. If the decompositionsubjacent to B is a decomposition into paths of G, then wesay that B is an `-decomposition into paths. We may omit thelength `, when it is clear from context.

(a) (b) (c)

(d) (e) (f)

Fig. 2: Examples of vanilla trails. The red edges are the initialand final edges.

Suppose there is a set F of edges of a graph G such thatG − F admits a decomposition D′ into paths. Moreover,suppose we can extend each path in D′ with one edgeof F at each of its end-vertices, and such that each edgeof F is used exactly once. In this case, each path of D′becomes a vanilla trail. Moreover, we can obtain a trackingdecomposition B of G. Now fix a vertex v of G. Thenumber h(v) of trackings of B that end immediately aftertraversing v is exactly the number of trails of D′ that have vas an end-vertex. When h(v) is large with respect to themaximum length of a tracking of B some properties arise.The following definition presents such a property that isfundamental in this subsection.

Definition 3.1 (Feasibility). Let ` be a positive integer. Let G bea graph and B an `-decomposition of G. We say that B is feasibleif for every v ∈ V (G) the following holds: if T is an `-trail of G(not necessarily admitting a tracking in B) and v is contained in

a cycle of T , then there an element B in B such that B containsan edge vu where u is an end-vertex of B and u /∈ V (T ).

Lemma 4.6 presents a sufficient condition for the fea-sibility of `-decompositions of graphs with restrictions onthe girth. Lemma 4.15 deals with the special case where thegraph is bipartite.

For each vanilla tracking B in G, let τ(B) be thenumber of end-vertices of B with degree greater than 1,and let B a feasible `-decomposition of G that minimizesτ(B) =

∑B∈B τ(B)

(in Figure 2a, we have τ(B) = 1, and

in Figures 2b-f, we have τ(B) = 2). If τ(B) = 0, then B is

an `-decomposition into paths. Thus, suppose that τ(B) > 0.Since τ(B) > 0, there is a vanilla tracking B0 in B that is nota path. Let x be an end-vertex of B0 with degree greaterthan 1 in B0, and let C be a cycle in B0 that contains x.Consider a neighbor v of x in C . Since B is feasible, thereis an element B1 in B that contains the edge vu, such thatu /∈ V (B0) and u is an end-vertex of B1. Now, let B′0 andB′1 such that B′0 = B0−vx+vu, B′1 = B1−vu+vx, and letB′ = B −B0 −B1 +B′0 +B′1. We have τ(B′0) = τ(B0)− 1.Suppose that the decomposition obtained B′ is feasible.If τ(B′1) ≤ τ(B1), then B′ is an `-decomposition of Gsuch that τ(B′) =

∑B∈B′ τ(B) <

∑B∈B τ(B) = τ(B),

a contradiction to the minimality of τ(B). Thus, we haveτ(B′1) = τ(B1) + 1 and B′1 contains a cycle C ′ thatcontains xv. Now, consider a neighbor v′ of x in C ′ suchthat v′ 6= v, and repeat this operation while necessaryconsidering B′1 and v′ instead of B0 and v. We can showthat, under some conditions, after repeating this operation afinite number of times, we obtain a better decomposition (an`-decomposition with more trackings that induce paths thanthe previous decomposition – see Lemma 3.2). In the com-plete version of this work, we formalize which propertiesthe sequence of trails must satisfy in order to guarantee thisimprovement. Here, we illustrate in Figure 3 a step-by-stepof the algorithm presented above.

3.2 Hanging edges and complete tracking decomposi-tionsAll concepts defined here refer to `-decompositions B of agraph G, for a positive integer `. Recall that any trackingin B has exactly two end-vertices, even if they coincide.For a tracking B in B, we denote by τ(B) the number ofend-vertices of B that have degree greater than 1 in B; anddefine τ(B) =

∑B∈B τ(B). Note that τ(B) ∈ {0, 1, 2}, and

τ(B) = 0 if and only if B is a path. Figure 2a illustrates thecases where τ(B) = 1, while Figures 2b–f illustrate the caseswhere τ(B) = 2.

Let uv be an edge of G, and let B be the element of Bthat contains uv. If B = x0x1 · · ·x` with x0 = u and x1 = v,or x` = u and x`−1 = v, then we say that uv is a pre-hanging edge at v in the decomposition B. If, moreover,dB(u) = 1, then we say that uv is a hanging edge at v inthe decomposition B. We denote by preHang(v,B)

(resp.

Hang(v,B))

the number of pre-hanging (resp. hanging)edges at v in the decomposition B. Note that every hangingedge at a vertex v is also a pre-hanging edge at v. Thus,we have preHang(v,B) ≥ Hang(v,B) for every vertex v.Let k be a positive integer. We say that B is k-pre-complete ifpreHang(v,B) > k for every v in V (G). If Hang(v,B) > k

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(a) (b)

(c) (d)

Fig. 3: Illustration of how to deal with a sequence of trailsobtained by the algorithm above. At each step, the dashededges are the edges that are switched.

for every v in V (G), then we say that B is k-complete. Wealso say that B is complete, if B is k-complete for some k.

Let J be the set of the edges of G that are initial or finaledges of trackings of B. Consider an orientation O of J withthe following property: if uv is an edge in J and B is theelement of B that contains uv, where B = x0x1 · · ·x` withx0 = u and x1 = v, or x` = u and x`−1 = v, then uv is di-rected (in O) from v to u. Note that d+

O(v) = preHang(v,B)for every vertex v ofG. Moreover, for every v inGwe defineB(v) := d−O(v). We note that if B is an `-decomposition of G,then

∑v∈V (G) B(v) = 2|B| = 2|E(G)|/`, because every

element of B has exactly two end-vertices (counted withmultiplicity). Now, we are ready to present the Disentan-gling Lemma.

Lemma 3.2 (Disengangling Lemma). Let ` be a positive integerand G be a graph. If B is a k-complete feasible `-decompositionof G and τ(B) > 0, then there is an `-decomposition B′ of G withthe following properties.

• τ(B′) < τ(B);• B′(v) = B(v) for every v ∈ V (G);• B′ is k-complete.

4 DECOMPOSITION OF REGULAR GRAPHSINTO PATHS OF FIXED LENGTH

In this section we apply the Disentangling Lemma presentedin Section 3 and obtain decompositions into paths of fixedlength of a large family of regular graphs.

Kotzig [31] and, independently, Bouchet and Fou-quet [14] proved that a 3-regular graph admits a P3-decomposition if and only if it contains a perfect matching.The proof of this result introduce the structure of the proofwe use in this section, and that we generalize for highlyedge-connected graphs in section 5. Let M be a perfect

matching in a 3-regular graphG. The subgraphG−M is a 2-factor of G. Choose an Eulerian orientation for G−M and,for each edge xy in M , let Pxy be the subgraph of G thatcontains xy and the edges of G−M that leave x and y withrespect to the orientation chosen. The set {Pxy : xy ∈M} isa P3-decomposition of G (see Figure 4).

Fig. 4: Illustration of the proof of Kotzig’s, and Bouchet andFouquet’s Theorem.

Kotzig asked what are the necessary and sufficient con-ditions for an odd `-regular graph G to be decomposablein paths of length `. A necessary condition is the existenceof an b`/2c-factor (and, consequently, an d`/2e-factor) in G.To see this, let D be a decomposition G into `-paths and,for each path P = x0x1 · · ·x` ∈ D, consider the subsetFP = {x1x2, x3x4, . . . , x`−2x`−1}; the subgraph of G givenby F = ∪P∈DFP is an b`/2c-factor of G, and the subgraphG− E(F ) is an d`/2e-factor of G (see Figure 5).

(a) (b)

Fig. 5: 5-regular graph decomposed (a) into copies of P5 and(b) into a 2-factor and a 3-factor.

Favaron, Genest, and Kouider [21] showed that theexistence of an b`/2c-factor is not sufficient for G to bedecomposable into paths of length ` (see Figure 6). On theother hand, they showed that, for a 5-regular graph to admita P5-decomposition, it is sufficient for it to contain a perfectmatching and no cycles of length 4, and stated the followingconjecture.

Conjecture 4.1 (Favaron–Genest–Kouider, 2010). Let G be an`-regular graph, ` odd. If G contains a perfect matching, then Gadmits a P`-decomposition.

Recently, we proved [12] that every triangle-free 5-regular graph that contains a perfect matching admits a P5-decomposition, thus verifying Conjecture 4.1 for 5-regulargraphs with girth at least 4. In Subsection ?? we presenta generalization of the result in [12]. More specifically, weprove (see Theorem 4.13) that odd `-regular graphs withgirth at least `− 1 and containing a perfect matching admitP`-decompositions.

Let D be a decomposition of a graph G into trails. Givena vertex v of G, we denote by D(v) the number of elementsofD that have v as an end-vertex. We say thatD is balanced ifD(u) = D(v) for every u, v ∈ V (G). Note that D(v) ≡ d(v)

6

Fig. 6: 5-regular graph that contains a 2-factor, but does notadmit a P5-decomposition.

(mod 2) for every v in V (G). The following fact will be usedfrequently.

Fact 4.2. Let `, m and n be positive integers.

(i) If ` is odd and D is a decomposition of an `-regular graphinto trails of length `, then D is balanced.

(ii) If nm is even andD is a balanced decomposition into trailsof length ` of an m`-regular graph G with n vertices, thenD(v) = m for every vertex v of G.

Heinrich, Liu and Yu [26] proved that if G is a 3m-regular graph that contains an m-factor, then G admits abalanced P3-decompositions

(in this case, D(v) = m for

every v in V (G)). Kouider and Lonc [32] proved that if G

is a 2`-regular graph with girth g ≥ (` + 3)/2, then Gadmits a balanced P`-decomposition. They also proved thatif ` is even, then every bipartite `-regular graph with girthg ≥ (` + 3)/2 admits a P`-decomposition. Moreover, theystated the following conjecture.

Conjecture 4.3 (Kouider–Lonc, 1999). Every 2`-regular graphadmits a balanced P`-decomposition.

Thus, to solve Conjecture 4.3, it remains to prove it forgraphs with girth g < (` + 3)/2. Kouider and Lonc [32]proved it for ` = 3. Conjecture 4.3 is a strengthening (forpaths) of Conjecture 1.2 for decompositions of 2`-regulargraphs into trees with ` edges. Recently, we proved [13] thatConjecture 4.3 holds for ` = 4, i.e., ifG is an 8-regular graph,then G admits a balanced P4-decomposition.

In this section, we consider the problem of obtainingbalanced P`-decompositions of m`-regular graphs, for apositive integer m. We propose the following conjecture,that consists of a generalization of Conjecture 4.1 (item (i))and an equivalent form of Conjecture 4.3 (item (ii)). Theequivalence between Conjecture 4.3 and item (ii) of Con-jecture 4.4 follows from Petersen’s 2-factorization Theorem(see Theorem 2.1).

Conjecture 4.4 (Botler–Mota–Oshiro–Wakabayashi, 2015).For m and ` positive integers, the following statements hold.

(i) If ` is odd, then every m`-regular graph that contains anm-factor admits a balanced P`-decomposition;

(ii) Every 2m`-regular graph admits a balanced P`-decomposition.

Note that item (ii) of Conjecture 4.4 does not hold ifinstead of 2m` we consider m` with m odd and ` even.Indeed, if G is an m`-regular graph with n vertices, then|E(G)| = nm`/2. If G admits a P`-decomposition D,then |D| = nm/2, hence n must be even. By Fact 4.2 (ii)D(v) = m for every v ∈ V (G), but, since D is a pathdecomposition and G is Eulerian, D(v) must be even forevery v ∈ V (G), a contradiction.

In this section we prove that Conjecture 4.4 (i) holds forgraphs with girth at least g such that m > 2b(`−2)/(g−2)c,and Conjecture 4.4 (ii) holds for graphs with girth at least gsuch that m > b(` − 2)/(g − 2)c. In particular, we proveConjecture 4.4 (i) for m, ` such that m ≥ 2` − 1, andConjecture 4.4 (ii) for m, ` such that m ≥ ` − 1. Ourmain results, Theorems 4.10 and 4.11, are as follows: let g, `and m be positive integers with g ≥ 3. (i) if ` is oddand m > 2b(`− 2)/(g − 2)c, then every m`-regular graphwith girth at least g and that contains an m-factor admitsa balanced P`-decomposition; (ii) if m > b(` − 2)/(g − 2)c,then every 2m`-regular graph with girth at least g admits abalanced P`-decomposition.

Note that, for ` odd, our results implies that, for everypositive integer m ≥ 3, if G is an m`-regular graph withgirth at least `−1 and containing anm-factor, thenG admitsa balanced P`-decomposition. The results in [12] may beeasily generalized for proving this statement for m = 1 (seeTheorem 4.13), and a theorem in [32] deals with the casem = 2. Furthermore, in the case G is a bipartite graph, weprove that item (i) of Conjecture 4.4 holds for every m > `,while item (ii) holds for every m > `/2 (see Subsection 4.3).

4.1 Completeness, feasibility and prescribed girth

In this subsection we give sufficient conditions to obtain ak-complete `-decomposition from a (k + r)-pre-complete `-decomposition (see Lemma 4.5), and to an `-decompositionto be feasible (see Lemma 4.6).

Lemma 4.5. Let g, k, ` and r be positive integers with g ≥ 3 andlet G be a graph with girth at least g. If r ≥ b(` − 2)/(g − 2)cand G admits a (k + r)-pre-complete `-decomposition B, then Gadmits a k-complete `-decomposition B′ such that B′(v) = B(v)for every vertex v of G.

Proof. Let g, k, `, r and G be as in the hypothesis. Let B bea (k + r)-pre-complete `-decomposition of G, and let B′be a (k + r)-pre-complete `-decomposition of G such thatB′(v) = B(v) for every vertex v of G, and that maximizes∑v∈V (G) Hang(v,B′). We claim that B′ is k-complete, i.e.,

Hang(v,B′) > k for every vertex v of G.Suppose, for a contradiction, that B′ is not k-complete.

Thus, there is a vertex u of G such that Hang(u,B′) ≤ k.Since B′ is (k+r)-pre-complete, preHang(u,B′) ≥ k+r+1.Thus, there is at least r + 1 pre-hanging edges at u thatare not hanging edges at u, say ux1, . . . , uxr+1. Let T1 =y0y1 · · · y` be the element of B′ that contains ux1, where,without loss of generality, y0 = x1 and y1 = u, and let

7

X = {x1, . . . , xr+1}. Let x′1, . . . , x′s be the vertices in X con-

tained in V (T1), ordered by distance to y1 in the sequencey1y2 · · · y`. Let `0 be the distance from y1 to x′1 in T1 − y0y1,and let `i be the distance from x′i to x′i+1 in T1 − y0y1, for0 < i ≤ s− 1 (see Figure 7). Since the girth of G is at least g,we have `0 ≥ g − 1, and `i ≥ g − 2 for 1 ≤ i ≤ s− 1. Sincex′1, . . . , x

′s are ordered by distance to y1 in T1 − y0y1, we

have `−1 ≥∑s−1i=0 `i ≥ g−1+(s−1)(g−2) = s(g−2)+1.

Thus, s ≤ (` − 2)/(g − 2), which implies that s ≤ r. Since|X| = r + 1 > s, there is at least a vertex in X , say xp, thatis not a vertex of T1.

Let Tp be the element of B′ that contains uxp. We cansuppose without loss of generality that Tp = z0z1 · · · z`,where z0 = xp and z1 = u = y1. Let T ′1 = xpy1 · · · y`and T ′p = x1z1 · · · z`. Note that T ′1 = T1 − ux1 + uxp andT ′p = Tp−uxp+ux1, and consider B′′ = B′−T1−Tp+T ′1+T ′p.It’s not hard no check that B′′(v) = B′(v) = B(v) forevery v in V (G). Since xp /∈ V (T1), we have dT ′1(xp) = 1,which implies that uxp is a hanging edge at u in thedecomposition B′′. Therefore,

∑v∈V (G) Hang(v,B′′) >∑

v∈V (G) Hang(v,B′), a contradiction to the maximality of∑v∈V (G) Hang(v,B′).

· · · · · · · · · · · · · · ·

`0 `1· · · · · ·

`s−1

u x′1 x′2 x′s−1 x′s

Fig. 7: Obtaining a k-complete `-decomposition from a (k +r)-pre-complete `-decomposition.

The following lemma give a sufficient condition for acomplete `-decomposition to be feasible.

Lemma 4.6. Let g, k and ` be positive integers with g ≥ 3 andlet G be a graph with girth at least g. If k ≥ b(` − 2)/(g − 2)cand B is a k-complete `-decomposition of G, then B is feasible.

Proof. Let g, k, `, G and B be as in the hypothesis.Fix v ∈ V (G), and let T be an `-trail of G that con-tains a cycle, say C , such that v ∈ V (C). Since B isk-complete, Hang(v,B) > k ≥ b(` − 2)/(g − 2)c. Letvw1, . . . , vwb(`−2)/(g−2)c+1 be hanging edges at v in thedecomposition B, and let W = {w1, . . . , wb(`−2)/(g−2)c+1}.We claim that there is wi ∈ W tal que wi /∈ V (T ). Split Tat v, obtaining two trails T1 and T2 of length c1 and c2,respectively. Denote by si the number of vertices of W in Ti.If W ⊆ V (T ), then we have |W | ≤ s1 + s2. Suppose thats1 ≥ 1. Let w′1, . . . , w

′s1 be the vertices of W in V (T1),

ordered by distance to v in T1. Let `0 be the distancein T1 from v to w′1, and `i be the distance in T1 from w′ito w′i+1, for i > 0. Since G has girth at least g, we haved0 ≥ g − 1, and `i ≥ g − 2, for 1 ≤ i ≤ s1 − 1. Thus,c1 ≥

∑s1−1i=0 `i ≥ g − 1 + (s1 − 1)(g − 2) = s1(g − 2) + 1.

Analogously, we obtain that if s2 ≥ 1, then c2 ≥ s2(g−2)+1.Thus, if s1, s2 ≥ 1, then s1 + s2 ≤ (` − 2)/(g − 2), acontradiction. Now, suppose that s1 = 0. Since v is aninternal vertex of T , we have ` − 1 ≥ c2 ≥ s2(g − 2) + 1,

which implies that s1 + s2 = s2 ≤ (` − 2)/(g − 2). Inboth cases we have s1 + s2 ≤ b(` − 2)/(g − 2)c. Since|W | ≥ b(` − 2)/(g − 2)c + 1, we have W 6⊆ V (T ) and,therefore, there is a vertex wi ∈ W that is not a vertexof T .

Now, we use the lemma above to prove the main result ofthis subsection, that allow us to obtain an `-decompositioninto paths from an `-decomposition b(` − 2)/(g − 2)c-complete of graphs with girth at least g.

Lemma 4.7. Let g, k and ` be positive integers with g ≥ 3 andlet G be a graph with girth at least g. If k ≥ b(` − 2)/(g − 2)cand B is a k-complete `-decomposition of G, then G admits a k-complete `-decomposition into paths B′ such that B(v) = B′(v)for every vertex v of G.

Proof. Let g, k, `, G and B be as in the hypothesisof the lemma. Let B be the set of all k-complete `-decompositions B′ of G such that B′(v) = B(v) for everyvertex v of G. From the hypothesis of the lemma, we haveB 6= ∅. Let τ∗ = min{τ(B′) : B′ ∈ B} and let Bmin be anelement of B such that τ(Bmin) = τ∗. If τ∗ = 0, then Bmin

is an `-decomposition into paths and the proof is finished.Thus, suppose that τ∗ > 0. By Lemma 4.6, Bmin is a feasi-ble a `-decomposition. Since τ(Bmin) > 0, by Lemma 3.2applied with k, `, G, and Bmin, there is a k-complete `-decomposition B′ of G such that τ(B′) < τ(Bmin) = τ∗

and B′(v) = B(v) for every vertex v of G. Therefore, B′ isan element of B with τ(B′) < τ(Bmin), a contradiction tothe minimality of τ∗.

4.2 Decompositions of regular graphs

The proof of our main results on `-decompositions follow byinduction in `. To obtain these proofs, we need first to extendthe concept of balanced decompositions to tracking decom-positions. Let B be a tracking decomposition of a graph G.Recall that we defined B(v), for v ∈ V (G), as the number ofedges of G incident to v that are initial edges of trackings inB that start at v, or final edges of trackings in B that end atv. Analogously to the definition of balanced decompositioninto trails, we say that B is balanced if B(u) = B(v) for everyu, v ∈ V (G). Note that B is balanced if and only if thedecomposition into trails subjacent to B is also balanced.

The two following results consist of special cases of thetheorems we prove next.

Theorem 4.8 (Heinrich–Liu–Yu [26]). Let m be a positiveinteger. If G is a 3m-regular graph that contains an m-factor,then G admits a balanced 3-decomposition into paths.

Proposition 4.9. Letm be a positive integer. IfG is a 4m-regulargraph, then G admits a balanced 2-decomposition into paths.

Proof. Let m and G be as in the statement. Consider anEulerian orientation of G. Since G is 4m-regular, we haved+(v) = d−(v) = 2m for every v ∈ V (G). For eachv ∈ V (G), we decompose the set of edges that leave v intompaths of length 2. Let B be the 2-decomposition composedby any tracking composta of each of these 2-paths, and notethat B(v) = d−(v) = 2m for every vertex v of V (G). Thisconcludes the proof.

8

The following theorem is our main result about decom-positions into paths of odd length.

Theorem 4.10. Let `, g and m be positive integers such that `is odd and g ≥ 3, and let G be an m`-regular graph with girthat least g and that contains an m-factor. If m > 2b(`− 2)/(g −2)c, then G admits a balanced `-decomposition into paths (and,consequently, a balanced P`-decomposition).

Proof. The proof follows by induction on `. By Theorem 4.8,the statement holds for ` = 3 and g ≥ 3. Fix ` ≥ 5 andsuppose that the statement holds for `− 2.

Let M be an m-factor of G. The graph G − E(M) ism(` − 1)-regular and, thus, by Theorem 2.1, admits a 2-factorization {F1, . . . , Fm(`−1)/2}. Let H be the union of mof these factors. Thus, H is a 2m-factor of G and G′ = G −E(H) is anm(`−2)-regular graph with girth at least g. Notethat, m > 2b(`−2)/(g−2)c ≥ 2b(`−4)/(g−2)c. Therefore,by the induction hypothesis, G′ admits a balanced (` − 2)-decomposition into paths B′.

We claim that B′(v) = m for every vertex v in V (G).Let |V (G)| = n. Note that G′ contains nm(` − 2)/2edges and, consequently, B′ contains nm/2 paths. Thus,∑v∈V (G) B′(v) = nm. Since B′ is balanced, B′(v) = m

for every vertex v of G (see Fact 4.2). Choose an Eulerianorientation for H . Note that d+

H(v) = m = B′(v). Therefore,for each tracking B′ = x1 · · ·x`−1 em B′ we can chooseedges x1x0 and x`−1x` of H that leave x1 and x`−1,respectively, and add them to B, obtaining the `-trackingB = x0x1 · · ·x`−1x`. Since d+

H(v) = m = B′(v), we cando this operation in such a way to use each edge of Hexactly once. Let B be the `-decomposition obtained. Notethat the edges x1x0 and x`−1x` are pre-hanging edges, res-pectively, at x1 and x`−1 in the decomposition B. Moreover,B(v) = d−H(v) = m for every vertex v in V (G), because Hhas an Eulerian orientation. Therefore, B is balanced. Letr = b(` − 2)/(g − 2)c and k = m − r − 1. Note thatk ≥ b(` − 2)/(g − 2)c. By the definition of pre-hangingedge, the edges of H that leave a fixed vertex v of G areprecisely the pre-hanging edges at v in the decomposition B;consequently, B is (m − 1)-pre-complete, i.e., (k + r)-pre-complete. By Lemma 4.5 applied with g, k, `, and r, thegraph G admits a balanced k-complete `-decomposition B∗.Since k ≥ b(` − 2)/(g − 2)c, by Lemma 4.7, G admits abalanced k-complete `-decomposition into paths.

The proof of Theorem 4.11 is similar to the proof of The-orem 4.10, and uses Proposition 4.9 instead of Theorem 4.8.

Theorem 4.11. Let `, m and g ≥ 3 be positive integers. If m >b(` − 2)/(g − 2)c, then every 2m`-regular graph with girth atleast g admits a balanced `-path tracking decomposition.

Let ` be an odd positive integer. If we consider an m`-regular graph G with girth at least `− 1, then Theorem 4.10guarantees that G admits a decomposition into paths oflength `, for every integer m ≥ 3. The next results showthat this result also holds for m ≤ 2. The case m = 2 is aconsequence of the following result obtained by Kouider eLonc [32].

Theorem 4.12 (Kouider–Lonc [32]). If G is a 2`-regular graphwith girth at least g such that ` ≤ 2g − 3, then G admits abalanced P`-decomposition.

For the case m = 1, we generalize the result givenin [12], que states that every triangle-free 5-regular graphthat contains a perfect matching admits a P5-decomposition.The proof is a straightforward generalization of the proofin [12], and will be presented in a complete version of thiswork.

Theorem 4.13. Let ` be an odd positive integer. If G is an `-regular graph with girth at least `− 1 and that contains a perfectmatching, then G admits an `-decomposition into paths (and,consequently, a P`-decomposition).

4.3 Decompositions of bipartite graphs

In this subsection we show that for bipartite graphs we canobtain better bounds for the degree of the vertices. For that,we use versions of Lemmas 4.5, 4.6, and 4.7 proved in [10].The following lemmas will also be used in Section 5.

Lemma 4.14. Let k, ` and r be positive integers, andG a bipartitegraph. If r ≥ b`/2c and G admits a (k + r)-pre-complete `-decomposition B, then G admits a k-complete `-decomposition B′such that B′(v) = B(v) for every vertex v of G.

Proof. Let k, `, r, G = (A,B;E) and B as in thestatement of the Lemma. Let B′ be a (k + r)-pre-complete `-decomposition of G such that B′(v) = B(v)for every vertex v of G, and that maximizes the sum∑v∈V (G) Hang(v,B′). We claim that B′ is k-complete, i.e.,

Hang(v,B′) > k for every vertex v of G. Suppose, for a con-tradiction, that B′ is not k-complete. Then, there is a vertex u(without loss of generality) in A such that Hang(u,B′) ≤ k.Thus, there is at least r + 1 pre-hanging edges at u thatare not hanging edges at u, say ux1, . . . , uxr+1. Let T1 =y0y1 · · · , y` be the element of B′ that contains ux1, where,without loss of generality, y0 = x1 and y1 = u. The verticesof T1 in B are y0, y2, . . . , y2b`/2c. Since y1y0 is not a hangingedge, y0 ∈ {y2, . . . , y2b`/2c}. Thus, the number of verticesof T1 in B is at most b`/2c. Since r ≥ b`/2c, there is at leastone pre-hanging edge that is not hanging, say uxi, at u in thedecomposition B′ such that xi /∈ V (T1). Let Ti = z0z1 · · · z`be the element of B′ that contains uxi, where z0 = xie z1 = u. Let T ′1 = z0y1y2 · · · y` and T ′i = y0z1z2 · · · z`, andput B′′ = B′ − T1 − Ti + T ′1 + T ′i . Note that the pre-hangingedges in B′′ are the same in B′, however, the edge uxi is ahanging edge in B′′. Thus, we conclude that B′′(v) = B′(v)for every vertex v of G. Moreover, we have Hang(u,B′′) ≥Hang(u,B′) + 1, and Hang(v,B′′) = Hang(v,B′) for eve-ry vertex v in V (G) − u. Thus,

∑v∈V (G) Hang(v,B′′) >∑

v∈V (G) Hang(v,B′), a contradiction to the maximalityof the sum

∑v∈V (G) Hang(v,B′). Therefore, B′ is k-

complete.

Lemma 4.15. Let ` and k be positive integers andG be a bipartitegraph. If k ≥ d`/2e and B is a k-complete `-decomposition of G,then B is feasible.

Proof. Let `, k, G and B as in the hypothesis of thelemma. Fix v ∈ V (G), and let T be an `-trail of Gcontaining a cycle C such that v ∈ V (C). Since B isk-complete, Hang(v,B) > k. Let vw1, . . . , vwk+1 be thehanging edges at v in the decomposition B. We claim thatthere is an index 1 ≤ i ≤ k + 1 such that wi /∈ V (T ).

9

Let W = {w1, . . . , wk+1}. Let G = (A,B;E) and suppose,without loss of generality, that v ∈ A. Since G is bipartite,W ⊂ B. Moreover, since T contains a cycle, T contains atmost ` vertices, hence |V (T ) ∩ B| ≤ d`/2e ≤ k. But since|W | = k+ 1, we conclude that there is a vertex w ∈W suchthat w /∈ V (T ).

The proof of the next lemma is analogous to the proof ofLemma 4.7, using Lemma 4.15 instead of Lemma 4.6.

Lemma 4.16. Let k, ` be positive integers and let G be a bipartitegraph. If k ≥ d`/2e and B is a k-complete `-decomposition of G,then G admits a k-complete `-decomposition into paths B′ suchthat B′(v) = B(v) for every vertex v of G.

The two following theorems are the main results of thissubsection. Again, the proof for paths of even length isanalogous to the proof for paths of odd length. Here weshow only the proof for decompositions into paths of oddlength.

Theorem 4.17. Let ` and m be positive integers such that ` isodd, and let G be a bipartite m`-regular graph. If m > `, then Gadmits a balanced `-decomposition into paths (and, consequently,a balanced P`-decomposition).

Theorem 4.18. Let ` and m be positive integers and let G be abipartite 2m`-regular graph. If m > `/2, then G admits a bal-anced `-decomposition into paths (and, consequently, a balancedP`-decomposition ).

Proof of Theorem 4.17. Since every bipartite regular graphadmits a decomposition into 1-factors, the statement holdsfor ` = 1. Fix ` ≥ 3 and suppose that the statement holdsfor `−2. Let {M1, . . . ,Mm`} be a 1-factorization ofG. LetHbe a graph obtained by the union of 2m of these 1-factors,and put G′ = G − E(H). The graph G′ is an m(` − 2)-regular graph. Since m > `/2 > (`− 2)/2, by the inductionhypothesis, G′ admits a balanced (`−2)-decomposition intopaths B′.

By Fact 4.2, we have B′(v) = m for every vertex vin V (G). Choose an Eulerian orientation to H . Note thatd+H(v) = m = B′(v). Thus, we can extend each tracking P

of B′ for a vanilla tracking T by adding to P an edge leav-ing each of its end-vertices. Let B be the `-decompositionobtained, and note that B(v) = d−H(v) = m for everyvertex v in V (G). Thus, B is balanced. Let r = (` − 1)/2and k = m − r − 1, and note that k ≥ (` + 1)/2. By thedefinition of pre-hanging edge, the edges of H that leavea fixed vertex v of G are precisely the pre-hanging edgesat v in the decomposition B; consequently, B is (m − 1)-pre-complete. By Lemma 4.14 applied with k, `, and r, thegraph G admits a balanced k-complete `-decomposition B∗.Since k ≥ (` + 1)/2, by Lemma 4.16, G admits a balancedk-complete `-decomposition into paths.

The following theorem summarizes the results presentedin this section.

Theorem 4.19. Let `, g and m be positive integers, where g ≥ 3.Then, the following statements hold.

(1) If ` is odd,

a) If m > 2b(`−2)/(g−2)c and G is an m`-regulargraph with girth at least g and containing an m-factor, then G admits a balanced P`-decomposition.

b) If g ≥ ` − 1 and G is an m`-regular graph withgirth at least g and containing an m-factor, then Gadmits a balanced P`-decomposition.

c) If m > `, and G is a bipartite m`-regular graph,then G admits a balanced P`-decomposition.

(2) If m > b(` − 2)/(g − 2)c and G is a 2m`-regulargraph with girth at least g, then G admits a balancedP`-decomposition.

(3) If m > `/2 and G is a bipartite 2m`-regular graph, thenG admits a balanced P`-decomposition.

5 HIGHLY EDGE-CONNECTED GRAPHS

In this section we study the problem of decomposing highlyedge-connected graphs into paths. More specifically, wedeal with the following conjecture proposed by Barat andThomassen [3].

Conjecture 5.1 (Barat–Thomassen, 2006). For each tree T ,there is a positive integer kT such that, if G is a kT -edge-connected graph and |E(G)| is divisible by |E(T )|, thenG admitsa T -decomposition.

Barat and Thomassen [3] proved that in the case where Tis the clawK1,3, this conjecture is equivalent to Tutte’s Weak3-flow Conjecture, posed by Jaeger [28]. They also noted thatConjecture 5.1 does not hold when T is a graph that containscycle. In a series of papers, Thomassen [46], [47], [48], [49],[50] verified Conjecture 5.1 for stars, some bistars, paths oflength 3, and paths of length 2k, for every positive integer k.We proved Conjecture 5.1 for paths of length 5 [9] and, morerecently, for paths of every fixed length [10]. Merker provedConjecture 5.1 for trees with diameter 3 and 4 [36], and sometrees with diameter 5, including the path of length 5 [45].Using a different approach, Bensmail, Harutyunyan, Le, andThomasse [6] also proved Conjecture 5.1 for paths of fixedlength. In this section, we present a simplification of theproof presented in [10].

First, we consider a result proved by Barat and Gerb-ner [2] and, independently, by Thomassen [49], that saysthat Conjecture 5.1 is equivalent to the following version forbipartite graphs.

Conjecture 5.2. For each tree T , there is a positive integer k′Tsuch that, if G is a bipartite k′T -edge-connected graph and |E(G)|is divisible by |E(T )|, then G admits a T -decomposition.

More specifically, these authors proved the followingtheorem.

Theorem 5.3 (Barat–Gerbner [2]; Thomassen [49]). Let T be atree with ` edges, ` > 3. The following statements are equivalent.

(i) There is a positive integer k′T such that, if G is a bipartitek′T -edge-connected graph and |E(G)| is divisible by `,then G admits a T -decomposition.

(ii) There is a positive integer kT such that, if G is a kT -edge-connected graph and |E(G)| is divisible by `, then Gadmits a T -decomposition.

Furthermore, kT ≤ 4k′T + 16`6`+1 and if T has diameter atmost 3, then kT ≤ 4k′T + 16`(`+ 1).

10

To prove Theorem 5.3, these authors proved the follow-ing result.

Lemma 5.4. Let T be a tree with ` edges, ` > 3, and let k be apositive integer. If G is

(4k + 16`6`+1

)-edge-connected, then G

can be decomposed into two graphs G1 and G2 such that

i) G1 is bipartite and k-edge-connected;ii) G2 admits a T -decomposition.

Lemma 5.4 says, in other words, that if G is highlyedge-connected, then we can remove copies of T from Gin such a way to obtain a bipartite graph that still ishighly edge-connected. Thus, from now one, we supposethat the graph G is bipartite. The main result of thissection, Corollary 5.35, says that if ` is a positive integerand G is a bipartite 2(13`+ 4r − 4)-edge-connected, wherer = max{32(` − 1), `(` + 1)}, and |E(G)| is divisible by `,then G admits a P`-decomposition.

As well as the proofs in Section 4, the proof of the mainresult of this section consists basically of a factorization stepand an induction step.

This section is organized as follows. In Subsection 5.1,we present some results that are used in the proofs ofSubsection 5.2. In Subsection 5.2, we adapt the definitionsof factors and factorizations for highly edge-connected bi-partite graphs, we define bifactorizations and prove a ver-sion of Petersen’s Theorem (Theorem 2.1) for highly edge-connected graphs. In Subsection 5.3, we adapt the definitionof balanced decomposition for graphs that admit bifactor-izations, and use Lemmas 4.14 and 4.16 to decompose thesegraphs into paths of fixed length. In Subsection 5.4, we puttogether the results of Subsections 5.2 and 5.3, and prove themain result of this section.

5.1 Preliminary resultsIn this subsection, we define two basic operations on graphs:vertex splitting (Subsection 5.1.1) and edge lifting (Subsec-tion 5.1.2). Furthermore, we present classical results that re-late this operations to edge-connectivity. In Subsection 5.1.3,we show some properties of highly edge-connected graphsthat are used in the proofs of Subsection 5.2.

5.1.1 Vertex SplittingLet G = (V,E) be a graph. Given a vertex v of G, a set Sv ={d1, . . . , dsv} with sv positive integers is called a subdegreesequence for v if d1+· · ·+dsv = dG(v). We say that a graphG′

is obtained from G by a (v, Sv)-splitting if G′ is composed ofG − v together with sv new vertices v1, . . . , vsv and dG(v)new edges such that dG′(vi) = di, for 1 ≤ i ≤ sv , and⋃svi=1NG′(vi) = NG(v).

For a given set V ′ = {v1, . . . , vr} with r vertices of G,let Sv1 , . . . , Svr be subdegree sequences for v1, . . . , vr , re-spectively. Let H1, . . . ,Hr be graphs obtained as follows:H1 is obtained from G by a (v1, Sv1)-splitting, the graphH2 is obtained from H1 by a (v2, Sv2)-splitting, and so on,until Hr , that is obtained from Hr−1 by a (vr, Svr )-splitting.We say that each Hi is an {Sv1 , . . . , Svi}-detachment of G.Roughly speaking, a detachment of G is a graph obtainedby successive applications of splittings operations in verticesof G. In Figure 8, the graph H is an {Sa, Se}-detachmentof G, where Sa = {2, 3} e Se = {2, 2, 2}. The next result

give us sufficient conditions for the existence of a 2k-edge-connected detachment of a 2k-edge-connected graph.

b a

ce

f

d g

G

ba1

a2

c

e1

e2 e3

f

d g

H

Fig. 8: A graph G and a graph H that is an {Sa, Se}-detachment of G.

Lemma 5.5 (Nash–Williams [39]). Let G be a 2k-edge-connected graph, with k ≥ 1 and V (G) = {v1, . . . , vn}. Foreach v ∈ V (G), let Sv = {dv1, . . . , dvsv} be a subdegree sequencefor v such that dvi ≥ 2k for i = 1, . . . , sv . Then, there is a2k-edge-connected {Sv1 , . . . , Svn}-detachment of G.

5.1.2 Edge liftingLet G = (V,E) be a graph and u, v, w distinct verticesof G such that uv, vw ∈ E. The multigraph G′ =

(V, (E \

{uv, vw})∪{uw})

is called an uw-lifting (or, simply, a lifting)at v (see Figure 9). Note that G′ may contain parallel edgesjoining u and w. If for every distinct pairs x, y ∈ V \{v}, themaximum number of edge-disjoint paths joining x to y inG′

is the same as in G, then the lifting at v is called admissible.If v is a vertex of degree 2, then the lifting at v is alwaysadmissible. Such a lifting, together with the removal of vis called a suppression of v. The next lemma is known asMader’s Lifting Theorem.

v

u

w

G

v

u

w

G

Fig. 9: A graph G and a graph H that is an uw-lifting of G.

Theorem 5.6 (Mader [34]). Let G be a multigraph and v avertex of G. If v is not a cut-vertex, dG(v) ≥ 4, and v has at leasttwo neighbors, then there is an admissible lifting at v.

The following lemma will be useful in the applicationof Mader’s Lifting Theorem. In what follows, we denoteby pG(x, y) the maximum number of pairwise edge-disjointpaths joining the vertices x to y in the graph G.

Lemma 5.7. Let G be a multigraph and k a positive integer. If vis a vertex in G such that d(v) < 2k and pG(x, y) ≥ k for anytwo distinct neighbors x and y of v, then v is not a cut-vertex.

Proof. Let G, k and v as in the statement. Suppose, fora contradiction that v is a cut-vertex, and let x be yneighbors of v in different components of G − v. Every

11

path joining x to y contains v as an internal vertex. SincepG(x, y) ≥ k, then there are P1, . . . , Pk pairwise edge-disjoint paths joining x to y. Since dPi(v) ≥ 2, it followsthat dG(v) ≥

∑ki=1 dPi(v) = 2k, a contradiction.

5.1.3 High edge-connectivityIn this subsection, we present some properties shown bygraphs with given edge-connectivity.

If G is a graph that contains 2k pairwise edge-disjointspanning trees, then, clearly, G is 2k-edge-connected. Theconverse is not true, but as stated in the next theorem, every2k-edge-connected graph contains k such trees.

Theorem 5.8 (Nash-Williams [37]; Tutte [51]). Let k be apositive integer. If G is a 2k-edge-connected graph, then Gcontains k pairwise edge-disjoint spanning trees.

The following recent result of Lovasz, Thomassen, Wuand Zhang [33] improves a result of Thomassen [48], and al-lows us, under obvious necessary conditions, and a minimaledge-connectivity, to find special orientations of the edges ofa graph.

Theorem 5.9 (Lovasz–Thomassen–Wu–Zhang [33]). Let k ≥3 a positive integer and G a (3k − 2)-edge-connected graph.Let p : V (G) → {0, . . . , k − 1} be such that

∑v∈V (G) p(v) ≡

|E(G)| (mod k). Then, there is an orientation O of G such thatd+O(v) ≡ p(v) (mod k), for every vertex v of G.

Combining the two results above, we prove the fol-lowing lemma, that allows us to deal with highly edge-connected bipartite graphs as regular graphs. This lemmais a straightforward generalization of Proposition 2 in [49].

Lemma 5.10. Let k ≥ 3 and r be positive integers. If G =(A1, A2;E) is a (6k + 4r − 4)-edge-connected bipartite graphand |E| is divisible by k, then G admits a decomposition intotwo r-edge-connected spanning graphs G1 and G2 such that, thedegree in Gi of each vertex of Ai is divisible by k, for i = 1, 2.

Proof. Let k, r and G = (A1, A2, E) be as in the statement ofthe lemma. By Theorem 5.8, G contains 3k+2r−2 pairwiseedge-disjoint spanning trees. Let H1 be the union of r ofthese trees, let H2 be the union of other r of these trees,and let H3 = G − E(H1) − E(H2). Clearly, H1 and H2 arer-edge-connected, and H3 is (3k − 2)-edge-connected. Letp : V (H3)→ {0, . . . , k−1} be such that p(v) ≡ (k−1)dH1

(v)(mod k) if v is a vertex of A1, and p(v) ≡ (k − 1)dH2

(v)(mod k) if v is a vertex of A2. Thus, the following holds,where the congruences are taken modulo k.∑

v∈V (G)

p(v) =∑v∈A1

p(v) +∑v∈A2

p(v)

≡ (k − 1)(|E(H1)|+ |E(H2)|)≡ (k − 1)(|E| − |E(H3)|)≡ k (|E| − |E(H3)|)− |E|+ |E(H3)|≡ |E(H3)|.

SinceH3 is a spanning (3k−2)-edge-connected subgraphof G, by Theorem 5.9 there is an orientation O of H3 suchthat d+

O(v) ≡ p(v) (mod k) for every v ∈ V (H3) = V (G).For i = 1, 2 let Gi be the graph Hi together with the edgesof H3 that leave Ai with respect to the orientation O (note

that E = E(G1)∪E(G2)). Thus, dGi(v) = dHi(v)+d+O(v) ≡

kdHi(v) ≡ 0 (mod k) for every vertex v in Ai, and, more-over, Gi is r-edge-connected (because contains Hi).

The following theorem is a generalization of a result ofPetersen [40] that says that every cubic graph with no cut-edge contains a perfect matching.

Theorem 5.11 (Von Baebler [52] (see also [1, Theorem 2.37])).Let r ≥ 2 be a positive integer, and G an (r − 1)-edge-connectedr-regular multigraph with an even number of vertices. Then Gcontains a 1-factor.

The following results are obtained by generalizing atechnique used by Barat and Gerbner [2], and will be usedin the proof of Lemma 5.14.

Theorem 5.12 (Theorem 20 in [19]). Letm be a positive integer.If G is an m-edge-connected graph, then G contains a spanningtree T such that dT (v) ≤ ddG(v)/me + 2 for every vertex vof G.

Corollary 5.13. Let m be a positive integer. If G is an m-edge-connected graph, then G contains a spanning tree T such thatdT (v) ≤ 4 dG(v)/m for every vertex v of G.

Proof. From the edge-connectivity of G, we have dG(v) ≥ mfor every vertex v of G. Combining this with Theorem 5.12,we conclude that G contains a spanning tree T such thatdT (v) ≤ ddG(v)/me+2 ≤ (dG(v)/m)+3 ≤ 4 dG(v)/m.

Lemma 5.14. Let k, m and r be positive integers, and let G =(A,B;E) be a bipartite graph. If G is 8md(k + r)/ke-edge-connected and, for every v ∈ A, dG(v) is divisible by k + r,then G admits a decomposition into two spanning graphs Gkand Gr such that Gk is m-edge-connected and, for every vertexv ∈ A, we have dGk(v) = k

k+rdG(v) and dGr (v) = rk+rdG(v).

Proof. Let k, m, r and G = (A,B;E) be as in the hypothesisof the lemma. Since G is 8md(k+ r)/ke-edge-connected, byTheorem 5.8 we conclude that G contains at least 4md(k +r)/ke pairwise edge-disjoint spanning trees. Now, partitionthe set of these 4md(k+r)/ke spanning trees into m set, sayT1, . . . , Tm, with 4d(k + r)/ke spanning trees each, and putGi =

⋃T∈Ti T , for i = 1, . . . ,m.

Clearly, Gi is 4d(k + r)/ke-edge-connected. By Corol-lary 5.13, Gi contains a spanning tree Ti such that, for everyv ∈ V (Gi),

dTi(v) ≤ 1

d(k + r)/kedGi(v) ≤

(k

k + r

)dGi(v).

Let G′ = ∪mi=1Ti. Clearly, G′ is m-edge-connected. Notethat, for every v ∈ V (G),

dG′(v) =m∑i=1

dTi(v) ≤(

k

k + r

) m∑i=1

dGi(v)

≤(

k

k + r

)dG(v).

Let Gk be the bipartite graph obtained from G′ byadding, for each vertex v in A, exactly

(k/(k + r)

)dG(v) −

dG′(v) edges of G − E(G′) that are incident to v (note that(k/(k + r)

)dG(v) is an integer). Therefore, every vertex

v ∈ A has degree exactly(k/(k + r)

)dG(v) in Gk. To

conclude the proof, put Gr = G− E(Gk).

12

5.2 FactorizationsThe main objective of this subsection is to show that somehighly edge-connected graphs admit “well-structured” de-compositions, called bifactorizations, which are importantstructures in the proof of the main theorem of this section(presented in subsection 5.4).

5.2.1 Fractional factorizationsBefore defining bifactorizations, we must generalize theconcepts of factors and factorizations. We extend the ideasdeveloped in [9] and formalize some ideas presented in [46].

Definition 5.15 (Factor). Let r and ` be positive integers andG = (V,E) be a graph. Let X ⊂ V and F ⊂ E. We saythat F is an (X, r, `)-factor of G if dF (v) = (r/`)dG(v), forevery v ∈ X .

Note that if G is an `-regular graph, then a(V (G), r, `

)-

factor is precisely an r-factor of G.

Definition 5.16 (Fractional factorization). Let ` and k bepositive integers such that ` − k is a positive even number. LetG = (V,E) be a graph and let X ⊂ V . We say that a partitionF = {M1, . . . ,Mk, F1, . . . , F(`−k)/2} of E is an (X, k, `)-fractional factorization of G if the following properties hold.

• Mi is an (X, 1, `)-factor of G, for 1 ≤ i ≤ k; and• Fj is an Eulerian (X, 2, `)-factor of G, for 1 ≤ j ≤

(`− k)/2.

Note that if G contains an (X, 1, `)-factor, then dG(v) isdivisible by ` for every v ∈ X . Thus, this fact implies that,if G admits an (X, k, `)-fractional factorization, then d(v) isdivisible by ` for every v ∈ X . The next lemma is the coreof this subsection.

Lemma 5.17. Let ` be an odd positive integer. If G = (A,B;E)is an (` − 1)-edge-connected bipartite graph such that dG(v)is divisible by ` for every v ∈ A, then G admits an (A, 1, `)-fractional factorization.

Proof. Let ` and G = (A,B;E) be as in the hypothesis. First,we want to apply Lemma 5.5 to obtain an (` − 1)-edge-connected graph G′ with maximum degree 2`− 3. For that,for each vertex v ∈ B, we choose integers sv ≥ 1 and 0 ≤rv < `−1 such that dG(v) = (`−1)sv+rv . Put dv1 = `−1+rvand dv2 = · · · = dvsv = ` − 1. Moreover, for every vertexv ∈ A, we put sv = dG(v)/` and dvi = ` for 1 ≤ i ≤sv . By Lemma 5.5

(applied with parameters ` − 1 and the

integers sv , dvi (1 ≤ i ≤ sv) for every v ∈ V (G)), there

is an (` − 1)-edge-connected bipartite graph G′ obtainedfrom G by splitting of each vertex v of A into sv verticesof degree `, and each vertex v of B into a vertex of degree` − 1 + rv < 2` − 2 and sv − 1 vertices of degree ` − 1.Let A′ and B′ the set of vertices of G′ obtained from thevertices of A and B, respectively. For ease of notation, ifv ∈ (A′ ∪ B′) \ (A ∪ B) we also denote by v the vertex inA ∪B that originated v.

The next step is to obtain an `-regular multigraph G∗

from G′ through lifting operations. For that, we add someedges to vertices in A′ and remove even degree verticesfrom B′ with successive applications of Mader’s LiftingTheorem as follows. Let G′0, G

′1, . . . , G

′λ be a maximal se-

quence of graphs such that G′0 = G′ and (for i ≥ 0)

G′i+1 is the graph obtained from G′i by the application ofan admissible lifting at an arbitrary vertex v ∈ B′ withdG′(v) /∈ {1, 2, `}.

Recall that, given any two distinct vertices of G′, say xand y, we denote by pG′(x, y) the maximum number ofpairwise edge-disjoint paths joining x to y in G′. We claimthat pG′i(x, y) ≥ ` − 1 for any x, y in A′ and every i ≥ 0.Clearly, pG′0(x, y) ≥ `−1 holds for any x, y inA′, becauseG′

is (` − 1)-edge-connected. Fix i ≥ 0 and suppose thatpG′i(x, y) ≥ ` − 1 holds for any x, y in A′. Let x, y twovertices in A′. Since G′i+1 is the graph obtained from G′i bythe application of an admissible lifting at a vertex v in B′,we have pG′i+1

(x, y) ≥ pG′i(x, y) ≥ `− 1.We claim that, if v ∈ B′, then dG′λ(v) ∈ {2, `}. Suppose,

for a contradiction, that there is a vertex v in B′ such thatdG′λ(v) /∈ {2, `}. Note that dG′i(u) ≥ dG′i+1

(u) ≥ 2 for everyu ∈ V (G′) and every 0 ≤ i ≤ λ. Since dG′(u) ≤ 2` − 3for every u ∈ V (G′), we have 2 ≤ dG′i(u) ≤ 2` − 3 forevery 0 ≤ i ≤ λ. Thus, 2 ≤ dG′λ(v) ≤ 2` − 3. SincedG′λ(v) ≤ 2` − 3, and for any two neighbors x and yof v we have pG′λ(x, y) ≥ ` − 1, by Lemma 5.7 it followsthat v is not a cut-vertex of G′λ. Thus, by Mader’s LiftingTheorem (Theorem 5.6) applied toG′λ, there is an admissiblelifting at v. Therefore, G′0, G

′1, . . . , G

′λ is not maximal, a

contradiction.In G′λ the set B′ may contain some vertices of degree 2.

For any such vertex v, if u and w are neighbors of v,we apply an uw-lifting at v, and remove the vertex v,i.e., we spireas v. Let G∗ be the graph obtained by theapplication of this process at every vertex of degree 2in B′. Note that the number of pairwise edge-disjoint pathsjoining two distinct vertices of A′ does not decrease, i.e.,pG∗(x, y) ≥ pGλ(x, y) ≥ ` − 1 for every x, y in A′. Clearly,the set of vertices of G∗ that belong to B′ is an independentset; we denote it by B∗ (eventually, B∗ = ∅). Furthermore,every vertex in B∗ has degree `.

Claim 5.18. G∗ is (`− 1)-edge-connected.

Proof. Let Y ⊂ V (G∗). Suppose that there is at least onevertex x ofA′ in Y and at least one vertex y ofA′ in V (G∗)−Y . Since there are at least `− 1 pairwise edge-disjoint pathsjoining x to y, there are at least ` − 1 edges with vertices inboth Y and V (G∗)− Y . Now, suppose A′ ⊂ Y (otherwise,A′ ⊂ V (G∗) − Y , and we put V (G∗) − Y instead of Y ),hence V (G∗) − Y ⊂ B∗. Since B∗ is an independent set,every edge with a vertex in V (G∗)− Y must have the othervertex in A′. Since every vertex in B∗ has degree `, there areat least ` edges with vertices in both Y and V (G∗)− Y .

We conclude that G∗ is an (` − 1)-edge-connected `-regular multigraph with vertex set A′ ∪ B∗, where B∗ isan independent set.

Since every vertex of G∗ has odd degree, |V (G∗)| iseven. By Theorem 5.11, G∗ contains a perfect matching M∗.Since the multigraph J∗ = G∗ − M∗ is (` − 1)-regular,Theorem 2.1 implies that J∗ admits a 2-factorization,say {F ∗1 , . . . , F ∗(`−1)/2}. Therefore, M∗, F ∗1 , . . . , F

∗(`−1)/2 is a

partition of E(G∗).Now, let’s go back to the bipartite graph G. Let xy be

an edge of G∗. If x ∈ A′ e y ∈ B∗, then xy correspondsto an edge of G. On the other hand, if x, y ∈ A′, then

13

there is a vertex vxy of B′ and two edges xvxy and vxyyin E(G′). Moreover, xy was obtained by a xy-lifting at vxy(either by the application of Mader’s Lifting Theorem orby the repression of vertices of degree 2). Then, each edgeof G∗ represents an edge of G or a 2-path in G such thatthe internal vertices of these 2-paths are always in B. Foreach edge xy ∈ E(G∗), define f(xy) = {xy} if x ∈ A′

and y ∈ B∗, and f(xy) = {xvxy, vxyy} if x, y ∈ A′. Notethat f(xy) ⊂ E(G) for each edge xy ∈ E(G∗). For a setS ⊂ E(G∗), put f(S) = ∪e∈Sf(e). The partition of E(G∗)in M∗, F ∗1 , . . . , F

∗(`−1)/2 induces a partition of E(G) into

M = f(M∗) and Fi = f(F ∗i ) for 1 ≤ i ≤ (`− 1)/2.We prove that {M,F1, . . . , F(`−1)/2} is an (A, 1, `)-

fractional factorization. Fix an index i ∈ {1, . . . , (` − 1)/2}.We show that M is an (A, 1, `)-factor of G and Fi is anEulerian (A, 2, `)-factor of G. Let v be a vertex of A in Gand put d′(v) = d(v)/`. Thus, we know that v is representedby d′(v) vertices inG∗. SinceM∗ is a perfect matching inG∗,there are d′(v) edges of M entering v and, since F ∗i is a2-factor in G∗, there are 2d′(v) edges of Fi incident to v.Finally, since F ∗i is Eulerian, the set Fi is Eulerian. Thisconcludes the proof.

Corollary 5.19. Let ` be an even positive integer. If G =(A,B;E) is a 16(`− 2)-edge-connected bipartite graph suchthat dG(v) is divisible by ` for every v ∈ A, then G admitsan (A, 2, `)-fractional factorization.

Proof. Let ` and G = (A,B;E) be as in the hypothesis. Weclaim that G contains an (A, 1, `)-factor F such that G − Fis (` − 2)-edge-connected (note that dG−F (v) is divisibleby ` − 1 for every v ∈ A). Since G is 16(` − 2) = 8(` −2)d`/(`− 1)e-edge-connected, by Lemma 5.14 (applied withparameters k = ` − 1, m = ` − 2 and r = 1), the graph Gadmits a decomposition into graphsGk andGr such thatGkis (` − 2)-edge-connected and dGk(v) =

((` − 1)/`

)dG(v),

and dGr (v) =(1/`)dG(v) for every v ∈ A. Thus, E(Gr) is

an (A, 1, `)-factor. By Lemma 5.17, Gk admits an (A, 1, ` −1)-fractional factorization F . Thus, since dGk(v) =

((` −

1)/`)dG(v) for every v ∈ A, we conclude that F ∪ {E(Gr)}

is an (A, 2, `)-fractional factorization of G.

5.2.2 Bifactorizations

To obtain a decomposition of highly edge-connected bi-partite graphs G into paths of fixed length `, we needto combine fractional factorizations. More specifically, wedecompose G into graphs G1 and G2 and then we combinea fractional factorization ofG1 with a fractional factorizationof G2. This process, called bifactorization, is defined asfollows.

Definition 5.20 (Bifactorization). Let ` and k be positiveintegers such that ` − k is a positive even number, and letG = (A1, A2;E) be a bipartite graph. Let F1,F2 families ofsubsets of E and Gi = G[∪F∈FiF ], for i = 1, 2. We say thatF = (F1,F2) is a (k, `)-bifactorization of G if the followingproperties hold.

(i) {G1, G2} is a decomposition of G; and(ii) Fi is an (Ai, k, `)-fractional factorization of Gi, for 1 ≤

i ≤ 2.

If G admits a (k, `)-bifactorization, we say that G is (k, `)-bifactorable.

The next concept will be used to guarantee that G1

and G2 have minimum degrees sufficiently high.

Definition 5.21 (Strong bifactorization). Let ` be an even posi-tive integer. LetG = (A1, A2;E) be a bipartite graph that admitsa (2, `)-bifactorization F = (F1,F2). Let Ei =

⋃F∈Fi F for

1 ≤ i ≤ 2. We say that F is strong if dEi(v) ≥ (`/2)(`/2 + 1)for every v in Ai for 1 ≤ i ≤ 2. If G admits a strong (2, `)-bifactorization, we say that G is strongly (2, `)-bifactorable.

For ease of notation, if F belongs to F1 or F2, then wesay that F is an element of F. In what follows, we givesufficient conditions for a bipartite graph to be stronglybifactorable.

Lemma 5.22. Let ` be an even positive integer. Let r =max{16(` − 2), (`/2)(`/2 + 1)}. If G is a (6` + 4r − 4)-edge-connected bipartite graph such that |E(G)| is divisible by `,then G is matronly (2, `)-bifactorable.

Proof. Let `, r e G = (A,B;E) be as in the hypothesis. ByLemma 5.10 (applied with ` and r), the graph G can be de-composed into two r-edge-connected spanning graphs G1

and G2 such that every vertex of A has degree divisibleby ` in G1, and every vertex of B has degree divisibleby ` in G2. But since r ≥ 16(` − 2), by Corollary 5.19(applied with `), we conclude that G1 admits an (A, 2, `)-fractional factorization and G2 admits a (B, 2, `)-fractionalfactorization. Thus, G is (2, `)-bifactorable. Since G1 and G2

are r-edge-connected, we have dG1(v) ≥ r ≥ (`/2)(`/2 + 1)

for every v ∈ A, and dG2(v) ≥ r ≥ (`/2)(`/2 + 1) for every

v ∈ B, from what we conclude that G is strongly (2, `)-bifactorable.

The result that will be used later is, specifically, thefollowing corollary.

Corollary 5.23. Let ` be a positive integer and let r =max{32(`−1), `(`+1)}. IfG is a (12`+4r−4)-edge-connectedbipartite graph such that |E(G)| is divisible by 2`, then G isstrongly (2, 2`)-bifactorable.

5.3 Decomposition of bifactorable graphs into `-pathsIn this subsection we prove that graphs that admit strongbifactorizations can be decomposed into paths of fixedlength. Here, we do not make use of high edge-connectivity.First, we adapt the concept of balanced decomposition ofa tricking decomposition for dealing with such graphs.Propositions 5.25 and 5.26 deal with the base cases of ourinduction step.

Definition 5.24 (Balanced tracking decompositions). Let `be a positive integer. Let G = (A,B;E) be a bipartite graphthat admits a

(2, 2`

)-bifactorization F = (F1,F2), and let Gi =

G[⋃

F∈Fi F]

for i = 1, 2. Let M1, N1 be the(A, 1, 2`

)-factors

of F, and let M2, N2 be the(B, 1, 2`

)-factors of F. We say that an

`-decomposition B of G is F-balanced if the following propertieshold.

• B(v) = dG1(v)/`+ dM2(v) + dN2(v), for every v ∈ A;and

• B(v) = dG2(v)/`+ dM1(v) + dN1(v), for every v ∈ B.

14

Our main goal is to prove Theorem 5.27, that guar-antees that it is possible to obtain an F-balanced `-complete `-decomposition into paths from a strong

(2, 2`

)-

bifactorization F. First, we show that from a (2, 4)-bifactorization (not necessarily strong) we can obtain abalanced 2-bifactorization into paths.

Proposition 5.25. If G is a bipartite graph that admits a (2, 4)-bifactorization F, then G admits an F-balanced 2-decompositioninto paths.

Proof. Let G = (A,B;E) and F be as in the statement.Let F = (F1,F2), where Fi = {Mi, Ni, Fi} for i = 1, 2.Let OFi be an Eulerian orientation of G[Fi], for i = 1, 2.Let C1 be the set of components of G[F1]. Let T be anelement of C1 and BT = a0b0a1b1 · · · asbsa0 a trackingof T , where ai ∈ A, and bi ∈ B, for 1 ≤ i ≤ s. Theset B′T = {aibiai+1 : 0 ≤ i ≤ s}, where as+1 = a0, is a2-decomposition of T in which every tracking has its end-vertices in A. Thus, B′1 = ∪T∈C1B′T is a 2-decompositionof G[F1] in which every tracking has its end-vertices in A.Analogously, G[F2] admits a 2-decomposition B′2 in whichevery tracking has its end-vertices in B.

Let v be a vertex of A. Since M1 and N1 are (A, 1, 4)-factors of G, we have dM1

(v) = dN1(v). Thus, the number

of edges inM1∪N1 incident to v is even, and we can decom-pose the edges inM1∪N1 incident to v into paths of length 2such that each path has its end-vertices in B. Choosing anytracking of these paths, we obtain a 2-decomposition B′′1of the edges in M1 ∪ N1 such that each path has its end-vertices in B. Analogously, there is a 2-decomposition B′′2 ofthe edges of M2 ∪ N2 such that each tracking has its end-vertices in A.

Let B = B′1 ∪ B′2 ∪ B′′1 ∪ B′′2 . Note that only the trackingsin B′1 and in B′′2 have end-vertices in A and, analogously,only the paths in B′2 and in B′′1 have end-vertices in B. Thus,if v is a vertex in A, then B(v) = B′1(v)+B′′2 (v) = dG(v)/2+dM2(v) + dN2(v), and if v is a vertex in B, then B(v) =B′2(v) + B′′2 (v) = dG(v)/2 + dM1(v) + dN1(v). Therefore, Bis an F -balanced 2-decomposition into paths of G.

The proof of the following result is similar to part ofthe proof given by Thomassen [46] for decomposition ofjiggly edge-connected graphs into paths of length 3, but herewe need to guarantee that the decomposition obtained isbalanced.

Proposition 5.26. If G is a bipartite graph that admits a (2, 6)-bifactorization F, then G admits an F -balanced 3-decompositioninto paths.

Proof. Let G = (A,B;E) be a bipartite graph that ad-mits a (2, 6)-bifactorization F = (F1,F2), where Fi ={Mi, Ni, Fi, Hi} for i = 1, 2. Let C1 be the set of com-ponents of G[F1 ∪ H1]. Let T be an element of C1 andBT = a0b0a1b1 · · · asbsa0 a tracking of T , where ai ∈ A andbi ∈ B, for 1 ≤ i ≤ s. The set B′T = {aibiai+1 : 0 ≤ i ≤ s},where as+1 = a0, is a 2-decomposition of T in which everytracking have its end-vertices in A. Thus, B′1 = ∪T∈C1B′T isa 2-decomposition of G[F1 ∪ H1] in which every trackinghave its end-vertices in A. Analogously, G[F2 ∪H2] admitsa 2-decomposition B′2 in which every tracking have its end-vertices in B.

LetGi = G[Mi∪Ni∪Fi∪Hi] for i = 1, 2. Note that, sinceM1 ∪ N1 is an (A, 2, 6)-factor and F1 ∪ H1 is an (A, 4, 6)-factor of G1, we have that dF1∪H1

(v) = (4/6)dG1(v) =

2dM1∪N1(v), for every vertex v in A. Note also that the

number of trackings in B′1 that finish at a vertex v (notethat we are not counting the trackings that start at verticesin A) equals 1

2dF1∪H1(v) = dM1∪N1

(v). Thus, we can extendeach tracking B of B′1 by adding an edge of M1 ∪ N1 atits final vertex, obtaining a 3-decomposition into paths B1

of G1. Analogously, we can extend each tracking B of B′2 byadding an edge of M2 ∪ N2 at its final vertex, obtaining a3-decomposition into paths B2 of G2.

Let B = B1 ∪ B2. If v is a vertex of A, then thenumber of trackings that have v as end-vertex is exactlydF1∪H1

(v)/2+dM2∪N2(v). Thus, we have B(v) = dF1

(v)/2+dH1

(v)/2+dM2(v)+dN2

(v) = dG1(v)/3+dM2

(v)+dN2(v).

Analogously, we have B(v) = dG2(v)/3 + dM1

(v) + dN1(v)

for every vertex v in B. Therefore, B is an F -balanced 3-decomposition into paths of G.

Now we are ready to prove the main result of thissection.

Theorem 5.27. Let ` be a positive integer. IfG is a bipartite graphthat admits a strong (2, 2`)-bifactorization F, then G admits anF -balanced `-decomposition into paths.

Proof. The proof follows by induction on `. By Proposi-tion 5.25, the statement holds for ` = 2; and by Propo-sition 5.26, the statement holds for ` = 3. Supposethat ` ≥ 4. Let G = (A1, A2;E) be a bipartite graphthat admits a strong (2, 2`)-factorizing F = (F1,F2).We claim that G admits an F -balanced `-pre-complete `-decomposition. Let F1 = {M1, N1, F1,1, . . . , F1,`−1} andF2 = {M2, N2, F2,1, . . . , F2,`−1}, and letGi = G

[⋃F∈Fi F

]for i = 1, 2.

From now on, fix i ∈ {1, 2}. Define d∗(v) = dGi(v)/(2`)for every vertex v ∈ Ai. Note that dFi,j (v) = 2d∗(v) =2dMi(v) = 2dNi(v) for every vertex v in Ai and 1 ≤ j ≤`−1. For j ∈ {`−2, `−1}, letOFi,j be an Eulerian orientationof G[Fi,j ]. Let Fi,j = F forw

i,j ∪ F backi,j , where F forw

i,j is the set ofedges of Fi,j leaving Ai with respect to OFi,j , and F back

i,j isthe set of edges of Fi,j entering Ai with respect to OFi,j . LetG′ = G−M1−N1−M2−N2−F forw

1,`−2−F forw1,`−1−F forw

2,`−2−F forw

2,`−1, and let F ′i = {F backi,`−2, F

backi,`−1, Fi,1, . . . , Fi,`−3}. Let

G′i = G[⋃

F∈F ′iF]. Note that G′i = Gi−Mi−Ni−F forw

i,`−2−F forwi,`−1. Then, for every v ∈ Ai, we have

dG′i(v) = dGi(v)−4d∗(v) = 2`d∗(v)−4d∗(v) = 2(`−2)d∗(v).(1)

Claim 5.28. F′ = (F ′1,F ′2) is a strong(2, 2(` − 2)

)-

bifactorization of G′.

Proof. To prove this claim, we must prove the following.

(i) F backi,`−2 and F back

i,`−1 are(Ai, 1, 2(`− 2)

)-factors of G′i;

(ii) Fi,j is an Eulerian(Ai, 2, 2(` − 2)

)-factor of G′i for

j = 1, . . . , `− 3;(iii) dG′i(v) ≥ (`− 2)(`− 1) for every vertex v ∈ Ai.

To prove items (i) and (ii), first note that, for everyv ∈ Ai, we have dF back

i,`−2(v) = dF back

i,`−1(v) = d∗(v) and

dFi,j (v) = 2d∗(v) for every 1 ≤ j ≤ ` − 3. From (1), we

15

conclude that F backi,`−2 and F back

i,`−1 are(Ai, 1, 2(` − 2)

)-factors

of G′i, and Fi,j is an(Ai, 2, 2(` − 2)

)-factor of G′i. Since F

is a (2, 2`)-bifactorization, F1,j and F2,j are Eulerian graphsfor 1 ≤ j ≤ `− 3.

It remains to prove (iii). Since dG′i(v) = 2(` − 2)d∗(v)e d∗(v) = dGi(v)/(2`) for every vertex v ∈ Ai, we havedG′i(v) = 2(`−2)

2` dG1(v) for every v ∈ Ai. Since F is a strong(

2, 2`)-bifactorization, we have dGi(v) ≥ `(` + 1) for every

v ∈ Ai. Thus, dG′i(v) ≥ (` − 2)(` + 1) > (` − 2)(` − 1) forevery v ∈ Ai.

Since F′ is a strong(2, 2(`− 2)

)-bifactorization of G′, by

the induction hypothesis, G′ admits an F′-balanced (`− 2)-decomposition into paths B′. Since B′ is an F′-balance’s (`−2)-decomposition into paths, we have

• B′(v) = dG′1(v)/(`− 2) + dF back2,`−2

(v) + dF back2,`−1

(v) forevery v ∈ A1;

• B′(v) = dG′2(v)/(`− 2) + dF back1,`−2

(v) + dF back1,`−1

(v) forevery v ∈ A2.

Now we want to extend each (`−2)-tracking of B′ to obtainan `-decomposition of G. For that, we add edges of E(G)−E(G′) at the end-vertices of the tracking of B′. For eachvertex v ∈ A1 (v ∈ A2), let Sv be the set of edges of M1 ∪N1∪F forw

2,`−2∪F forw2,`−1

(M2∪N2∪F forw

1,`−2∪F forw1,`−1

)incident to v.

Note that for each edge e in E(G) − E(G′) there is exactlyone vertex v ∈ V (G) such that e ∈ Sv . Then,

⋃v∈V (G) Sv =

E(G)−E(G′). Thus, if we prove that B′(v) = |Sv| for everyvertex v ∈ V (G), then we can extend each tracking B in B′by adding one edge at each of its end-vertices B.

Claim 5.29. B′(v) = |Sv|, for every v ∈ V (G).

Proof. First, note that, since F2,`−2 and F2,`−1 are Eulerian,we have dF back

2,`−2(v) = dF forw

2,`−2(v) and dF back

2,`−1(v) = dF forw

2,`−1(v)

for every vertex v in A1, and since F1,`−2 and F1,`−1 areEulerian, we have dF back

1,`−2(v) = dF forw

1,`−2(v) and dF back

1,`−1(v) =

dF forw1,`−1

(v) for every vertex v in A2.For every v ∈ Ai and every 1 ≤ j ≤ ` − 3, we have

dG′i(v)/(2(` − 2)) = dFi,j (v)/2 = d∗(v). Now, recall thatfor each vertex v ∈ Ai, we have dMi

(v) = dNi(v) = d∗(v).Thus, for each v ∈ A1, we have

B′(v) = dG′1(v)/(`− 2) + dF back2,`−2

(v) + dF back2,`−1

(v)

= 2d∗(v) + dF forw2,`−2

(v) + dF forw2,`−1

(v)

= dM1(v) + dN1

(v) + dF forw2,`−2

(v) + dF forw2,`−1

(v)

= |Sv|.

Similarly, we have B′(v) = |Sv| for each v ∈ A2.

We showed that every tracking B of B′ can be extendedby adding an edge at each of its end-vertices. Let B be thetracking decomposition obtained after this extensions. Weconclude that B is an `-decomposition of G.

Claim 5.30. B is F-balanced.

Proof. Let x0 be a vertex of A1. First, we show that B(x0) ≤dF forw

1,`−2(x0) + dF forw

1,`−1(x0) + dM2(x0) + dN2(x0). If there is

no tracking T = x0x1 · · ·x` in B, where x0x1 is an edge ofE(G) − E(G′), then B(x0) = 0. For each such tracking T ,by the construction of B, we know that x0x1 is an element

of Sx1. Since x1 is a vertex of A2, we have Sx1

⊂M2 ∪N2 ∪F forw

1,`−2 ∪ F forw1,`−1. Thus,

B(x0) ≤ dF forw1,`−2

(x0) + dF forw1,`−2

(x0) + dM2(x0) + dN2

(x0).

Now, we prove that B(x0) ≥ dF forw1,`−2

(x0) + dF forw1,`−1

(x0) +

dM2(x0)+dN2(x0). Note that if x0x1 is an edge of M2∪N2∪F forw

1,`−2∪F forw1,`−1 incident to x0 in A1

(these are the only edges

in G that can contribute to B(x0)), then, by the construction

of B, there is a tracking Q′ = x1 · · ·x` of a path such that thetracking Q = x0x1 · · ·x`x0 (of a vanilla trail) belongs to B.Thus, B(x0) = dF forw

1,`−2(x0)+dF forw

1,`−1(x0)+dM2

(x0)+dN2(x0).

Thus, for every vertex v ∈ A1 we have

B(v) = |F forw1,`−2(v)|+ |F forw

1,`−1(v)|+ |M2(v)|+ |N2(v)|= 2d∗(v) + dM2(v) + dN2(v)

= dG1(v)/`+ dM2(v) + dN2(v).

Analogously, we have B(v) = dG2(v)/` + dM1

(v) + dN1(v)

for each vertex v ∈ A2. Thus, B is an F -balanced `-decomposition.

Claim 5.31. B is `-pre-complete.

Proof. Let v ∈ Ai. We will show that preHang(v,B) > `.Note that, by the construction of B, the set of pre-hangingedges at v in the decomposition B is exactly Sv . Then,preHang(v,B) = |Sv| = B′(v). Since B′ is F-balanced,B′(v) ≥ dG′i(v)/(`− 2). Thus,

preHang(v,B) = B′(v) ≥ dG′i(v)/(`− 2)

= 2d∗(v) = dGi(v)/`.

Since F is a strong(2, 2`

)-bifactorization of G, we

have dGi(v) ≥ `(` + 1), from what we conclude thatpreHang(v,B) ≥ ` + 1. Therefore, B is a `-pre-complete`-decomposition.

Now we are able to conclude the proof. Put k = d`/2eand r = b`/2c. Note that ` = k+r. By Lemma 4.14 with k, `,and r, G admits a k-complete `-decomposition B′′ such thatB(v) = B′′(v) for every vertex v of G. Since B′′(v) = B(v)for every vertex v of G, B′′ is F-balanced. By Lemma 4.16, Gadmits a k-complete `-decomposition into paths such thatB∗(v) = B′′(v) for every vertex v of G. Therefore, G admitsan F -balanced `-decomposition into paths.

5.4 Decomposition of highly edge-connected graphsinto `-paths

In this section we put together the results of Section 5.2 andTheorem 5.27, and prove Conjecture 5.2 for paths of fixedlength. First, we prove that Conjecture 5.2 is equivalent tothe following conjecture.

Conjecture 5.32. For every tree T , there is a positive integer k′′Tsuch that, if G is a bipartite k′′T -edge-connected graph and |E(G)|is divisible by 2|E(T )|, then G admits a T -decomposition.

We will prove Conjecture 5.32 for the case where Tis a path. The following result shows the equivalence ofConjecture 5.32 and Conjecture 5.2.

Theorem 5.33. Let T be a tree with ` edges, ` ≥ 3, and let k bea positive integer. If G is a 2(k + `)-edge-connected graph such

16

that |E(G)| is divisible by |E(T )|, then there is a subgraph H ofG such that

• H admits a T -decomposition; and• G′ = G − E(H) is k-edge-connected and |E(G′)| is

divisible by 2|E(T )|.

Proof. Let G be as in the statement. If |E(G)| is divisibleby 2|E(T )|, then we put H as the empty graph Otherwise,|E(G)| − ` is divisible by 2|E(T )|. In this case, let H bea copy of T in G. By theorem 5.8, G contains k + ` edge-disjoint spanning trees. Since T has ` edges, H interceptsat most ` of these trees. Thus, G− E(H) contains at least kedge-disjoint spanning trees. Therefore,G−E(H) is k-edge-connected.

Theorem 5.34. Let ` be a positive integer, and let r =max{32(`−1), `(`+1)}. IfG is a (12`+4r−4)-edge-connectedbipartite graph such that |E(G)| is divisible by 2`, then G admitsa P`-decomposition.

Proof. Let `, r, and G be as in the hypothesis. By Corol-lary 5.23, G admits a strong (2, 2`)-bifactorization F. ByTheorem 5.27, G admits an F -balanced `-decompositioninto paths. Therefore, G admits a P`-decomposition.

Corollary 5.35. Let ` a positive integer, and let r = max{32(`−1), `(`+ 1)}. If G is a 2(13`+ 4r− 4)-edge-connected bipartitegraph such that |E(G)| is divisible by `, then G admits a P`-decomposition.

Proof. The proof follows by applying Theorem 5.33 with k =12`+ 4r − r and Theorem 5.34.

Corollary 5.36. Let ` be a positive integer, r = max{32(` −1), `(` + 1)}, and put k′T = 2(13` + 4r − 4). If G is a(4k′T + 16`6`+1)-edge-connected graph such that |E(G)| isdivisible by a`, then G admits a P`-decomposition.

Proof. The proof follows directly from Corollary 5.35 andTheorem 5.3.

6 CONCLUDING REMARKS

Graph decomposition is a topic that has shown to be richin conjectures and challenging problems that have broughtsignificant contributions to structural graph theory. In thiswork we developed a technique to deal with decomposi-tions of graphs into paths that has shown to be useful todeal with well-studied problems (Conjectures 1.2, 4.1, and5.1). Furthermore, the tools developed in this work haveled us to other new results as in [13], obtained during thewriting of this text.

We emphasize that this work has benefited greatly fromThomassen’s results for decompositions of highly edge-connected graphs; we hope that this connection has becomeclear to the reader familiar with this results. We also want tomention that the result obtained by Merker [36] contributesto the literature as an alternative to the factorization andbifactorization results presented in this work. In particular,if it is possible to generalize the Disentangling Lemma todeal with more general trees, Merker’s result can be appliedto solve Conjecture 5.1 for such trees.

While we were writing the main result of Section 5in [10], we were informed that Bensmail, Harutyunyan, Le,

and Thomasse [6] obtained a similar result using a differentapproach. Recently, together with Merker, these authors [5]proved Conjecture 5.1. This shows that the study of graphdecompositions allows us to explore many different tech-niques that help finding partial or complete solutions toopen problems, bringing nice contributions to structuralgraph theory. To illustrate this fact, we recommend [15], [23].

We plan to continue working on Conjectures 1.2 and 5.1,and try to extend the family of trees for which they hold.It would be very interesting to generalize the DisentanglingLemma to deal with more general structures, because, asobserved above, this would lead to a constructive proof ofConjecture 5.1. Furthermore, such a generalization wouldallow us to obtain results analogous to the ones in Section 4for more general classes of trees.

In another direction, we believe that it is possible toimprove the girth condition of Conjecture 4.1.

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