Sparsely intersecting perfect matchings in cubic graphs

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Combinatorica 34 (1) (2014) 61–94DOI: 10.1007/s00493-014-2550-4

SPARSELY INTERSECTING PERFECT MATCHINGSIN CUBIC GRAPHS

EDITA MACAJOVA, MARTIN SKOVIERA

Received October 20, 2008

In 1971, Fulkerson made a conjecture that every bridgeless cubic graph contains a familyof six perfect matchings such that each edge belongs to exactly two of them; equivalently,such that no three of the matchings have an edge in common. In 1994, Fan and Raspaudproposed a weaker conjecture which requires only three perfect matchings with no edgein common. In this paper we discuss these and other related conjectures and make astep towards Fulkerson’s conjecture by proving the following result: Every bridgeless cubicgraph which has a 2-factor with at most two odd circuits contains three perfect matchingswith no edge in common.

1. Introduction

Matching structures in cubic graphs have enjoyed continuous interest ingraph theory. The first notable result in this area was the celebrated Pe-tersen Matching Theorem [8] from 1891 which states that every bridgelesscubic graph has a perfect matching. This theorem has been followed by a se-ries of other results and conjectures that involve matchings in cubic graphs.One of such conjectures was proposed by Fulkerson [3] in 1971 in a paper onmathematical programming (actually, the conjecture was previously statedby C. Berge but not published; see [9]). Currently, this conjecture is consid-ered to be among the most challenging open problems in graph theory.

Conjecture 1.1 (Berge-Fulkerson). Every bridgeless cubic graph con-tains a family of six perfect matchings such that each edge is contained inexactly two of them.

Mathematics Subject Classification (2000): 05C15, 05C70

http://dx.doi.org/10.1007/s00493-014-2550-4

62 EDITA MACAJOVA, MARTIN SKOVIERA

If we regard perfect matchings as 0,1-functions on the set of edges,the above conjecture states that in every bridgeless cubic graph the con-stant function 2 can be obtained by adding several such functions together.In 1977, Seymour [9] showed that if subtraction is permitted, then 2 canindeed be so obtained. However, besides this result, no significant progresstowards Fulkerson’s conjecture has been made even if one restricts the scopeto narrower classes of graphs. In this situation it is natural to turn attentionto weaker problems with the hope that these might shed some more lighton the original problem. In 1994, therefore, Fan and Raspaud [2] made thefollowing conjecture.

Conjecture 1.2 (Fan and Raspaud). Every bridgeless cubic graph con-tains three perfect matchings with empty intersection.

Observe that if a cubic graph admits a family of six perfect matchingscovering each edge twice, then the intersection of any three of them is empty;hence, the validity of Conjecture 1.1 implies that of Conjecture 1.2. Nonethe-less, even this weaker conjecture remains basically unexplored.

To compare the relative strength of Conjecture 1.1 and Conjecture 1.2 itis convenient to consider a family F of k perfect matchings in a bridgelesscubic graph G such that any three of the matchings have empty intersection.Since G is cubic, k does not exceed six, and if k equals six, then F coverseach edge of G twice. It follows that the above two conjectures delimit aspectrum of four conjectures with Conjecture 1.1 being the strongest andConjecture 1.2 being the weakest of the four.

Conjecture 1.3 (k-Perfect-Matching Conjecture). Let k be an integerwith 3≤ k≤ 6. Every bridgeless cubic graph contains a family of k perfectmatchings such that any three of them have empty intersection.

It may be useful to note that the 5-Perfect-Matching Conjecture is equiv-alent to the 6-Perfect-Matching Conjecture, that is, to the Fulkerson Con-jecture. Consequently, there remain only three conjectures in this spectrum,all trivially satisfied for 3-edge-colourable graphs.

Conjecture 1.2 has recently reappeared in the context of Fano colour-ings [7], a geometrically motivated generalisation of the standard 3-edge-colourings. A Fano colouring is an edge-colouring of a cubic graph by pointsof the Fano plane such that the colours of any three edges meeting at a ver-tex form a line. A k-line Fano colouring is one that uses at most k differentlines as colour patterns around the vertices. Clearly, a graph admits a 1-lineFano colouring if and only if it is 3-edge-colourable, and the same holds truefor 2-line colourings and 3-line colourings. In [7] the authors showed that

SPARSELY INTERSECTING PERFECT MATCHINGS 63

every bridgeless cubic graph admits a 6-line Fano colouring and conjecturedthat four lines would do.

Conjecture 1.4 (4-Line Conjecture). Every bridgeless cubic graph hasa 4-line Fano colouring.

As was shown in [1, Theorem 3.1], a cubic graph admits a k-line Fanocolouring for some k ∈ 4,5,6 if and only if it has three parity subgraphswith no edge in common such that 7− k of them are perfect matchings.In particular, Conjecture 1.4 is equivalent to the 3-Perfect-Matching Con-jecture (see [7]). The geometric point of view, however, suggests an obviousrelaxation of the conjecture involving five lines of the Fano plane rather thanfour [7]. Unfortunately, even this weaker statement is not known to be true.

Conjecture 1.5 (5-Line Conjecture). Every bridgeless cubic graph hasa 5-line Fano colouring.

Recently, Kaiser and Raspaud [5,6] have verified Conjecture 1.5 forbridgeless cubic graphs of oddness two. Recall that the oddness of a bridge-less cubic graph G is the minimum number of odd circuits in a 2-factor ofG. Since oddness is always an even integer and oddness zero means that thegraph is 3-edge-colourable, cubic graphs of oddness two constitute a naturalclass of graphs with structure close to that of 3-edge-colourable graphs.

In the present paper we improve the result of Kaiser and Raspaud andmake a further step towards the Fulkerson conjecture by proving the follow-ing result.

Main Theorem. Every bridgeless cubic graph of oddness at most 2 con-tains three perfect matchings with empty intersection.

Our paper is organised as follows. The next section presents a brief surveyof basic terminology and notation used later in the paper. Section 3 intro-duces the fundamental concepts of the proof of Main Theorem and outlinesits main ideas. The next five sections are devoted to various stages of theproof. They contain a number further technical concepts and auxiliary re-sults that together imply the main result. The final proof concludes the lastsection of this paper.

2. Preliminaries

Throughout we consider graphs which may have multiple edges and loops.If G is a graph and H and K are subgraphs of G, we let H −K denote


the subgraph of H obtained by the removal of the edges of K (that is,H −K = H −E(K) in the standard notation) and by [H]G the subgraphinduced by the vertices of H.

A circuit in G is a connected 2-valent subgraph of G. An edge-cut, orsimply a cut, in G is a set X of edges for which there exists a partitionV1,V2 of the vertex-set of G such that X coincides with the set of alledges having exactly one vertex in each of V1 and V2. In other words, a cutis the set of edges that radiates from a set of vertices. If K1 and K2 aresubgraphs of G such that V (K1) is contained in one of the sets V1 and V2while V (K2) is contained in the other set, then the cut X is said to separateK1 from K2, or to be a K1-K2-cut.

Throughout this paper we make extensive use of various types of walks,paths, and their segments. The rest of this section is therefore devoted toestablishing this terminology. A walk in a graph G is a non-empty sequenceW = v0e1v1e2v2 . . .vk−1ekvk whose terms are alternately vertices and edgessuch that, for 1≤ i≤k, the edge ei joins the vertex vi−1 to the vertex vi. Awalk is closed if vk =v0. If no confusion arises, we encode W by its vertex-sequence v0v1 . . .vk−1vk or by its edge-sequence e1e2 . . .ek and say that W isa v0-vk-walk or an e1-ek-walk. The vertices v0 and vk are called, respectively,the initial and the terminal vertex of W , while e1 and ek are the first andthe last edge of W . The vertices v0 and vk and the edges e1 and ek arealso referred to as outer elements; other vertices and edges W are its innerelements. As usual, a path is a walk with no repetition of vertices and edges.

If K1 and K2 are subgraphs of G, then by a K1-K2-walk we mean a v1-v2-walk such that its initial vertex v1 belongs to K1, its terminal vertex v2belongs to K2 and all other vertices lie outside K1∪K2. Combined notationwill also be used: if K is a subgraph and e is an edge, then a K-e-walk is awalk whose initial vertex is the only vertex in K and whose last edge is e. Ane-K-walk is defined analogously. Finally, if S and T are sets whose elementsare either subgraphs or edges of G, then a S-T -walk is an X-Y -walk forsome X∈S and Y ∈T .

Each walk is understood to be directed from the initial to the terminalvertex. The walk obtained from W by reversing its direction will be denotedby W−1. Occasionally we apply this notation to an edge-encoded path oflength 1; in this case e−1 is meant to carry the orientation opposite to theone previously mentioned. If W is a u-v-walk and W ′ is a v-w-walk, then theu-w-walk obtained by concatenating W and W ′ will be denoted by WW ′.This operation is clearly associative.

Let W be a walk, and let e1 and e2 be two edges in the order of theirappearance on W . Then W [e1,e2] will denote the segment of W whose first


edge is e1 and last edge is e2. Furthermore,W [∗,e2] will be the initial segmentof W up to the edge e2 and W [e1,∗] will be the terminal segment of W frome2 up to the terminal vertex. We also employ the self-explanatory notation ofwalk segments where a bracket denotes a “closed end” and a parenthesis an“open end” of a segment. For example, W [e1,e2) means the segment whosefirst edge is e1 and whose last edge is the immediate predecessor of e2.

3. The method

Suppose that we are given a bridgeless cubic graph G and want to find threeperfect matchings F0, F1, and F2 in G such that F0 ∩F1 ∩F2 = ∅. If G is3-edge-colourable, then the colour classes of any 3-edge-colouring providethe required perfect matchings. If not, by Petersen’s Theorem G containsa perfect matching, say F . We split every even circuit of G−F into twoperfect matchings to obtain a pair of disjoint matchings M and N . It seemsnatural to take F , M , and N as a point of departure for constructing thedesired triple of perfect matchings in G. We now develop this idea in detail.

Let G be a cubic graph containing a 1-factor F , and let C be the 2-factorcomplementary to F ; note that we do not require G to be bridgeless. Wepartition C into its odd part A, consisting of all odd circuits of C, and its evenpart B, consisting of all even circuits. The odd circuits of C will be usuallydenoted by A1,A2, etc. In the even part B we choose a perfect matchingM and form an ordered triple M= (G,F,M); this triple is a fundamentalconcept of our proof. We call it a mesh on G. The subgraphs F , C=G−F ,and M will be called the 1-factor, the 2-factor, and the cofactor of M,respectively.

We say that a meshM is connected or bridgeless if its supporting graphG is connected or bridgeless, respectively. If G is disconnected, then eachcomponent G′ of G gives rise to a connected mesh M′ = (G′,F ′,M ′), acomponent of M, where F ′ = F ∩G′ and M ′ =M ∩G′. A connected meshwith exactly k odd circuits will be called a k-mesh; clearly, k must be even.

Before being able to use meshes as an effective device, we need to describea number of structures residing inside each mesh. A circuit of M is simplyany circuit of C. Edges not in C are of two types: a ray is an edge of F withend-vertices in different circuits ofM whereas a chord is an edge of F withend-vertices in the same circuit Z ofM. A ray incident with Z is said to bea ray of Z.

Many important features of meshes can be described by means of varioustypes of walks in the supporting graph. A walk in a mesh M= (G,F,M)is understood to be any walk whose edges are contained in F ∪B; no other


walks will be considered in this paper. An alternating walk is one in whichedges of M and those not in M alternate. A path in M is a path whoseedges are contained in F ∪B.

According to these definitions, walks may have their end-vertices in A,but they cannot have edges in A. A walk with end-vertices in different cir-cuits of A will be called transversal, otherwise it will be called local. A lo-cal walk which is completely contained within the subgraph [Z]G for somecircuit Z of M will be called stationary. Two walks in M are said to beindependent, or one of them is said to be independent of the other, if noedge of F occurs in both of them; they will be called internally independentif the walks resulting from severing the outer edges from both walks areindependent.

The second crucial concept of our proof is that of a chain. A chain in ameshM is an alternating walk without repeated edges which is either closedor its outer edges are rays. A trivial chain consists of a single F -edge.

There are two basic species of non-trivial chains. The first of them com-prises all even circuits of M, while the second one arises from componentsthe subgraph F∪M . Since F is a 1-factor and M is a matching of G disjointfrom F , we see that F ∪M is a spanning subgraph with each componentbeing a chain. Every chain that corresponds to a component of F ∪M willbe called a wire of M. Of course, there exist chains that are neither evencircuits nor wires of M, but each chain consists of pieces of even circuitsand wires linked together.

Next we examine intersections of chains and circuits in a mesh M. LetX be a chain intersecting a circuit Z of M and let Y be a component ofX∩[Z]G. Since Y is alternating, there is a unique minimal subchain Y of Xcontaining Y . We call Y a fragment of X on Z. We also define a cofragmentof X on Z to be any component of the spanning subgraph Z−X of Z. If Zis odd, a chain can have at most two fragments on it, each of the fragmentsbeing a trivial chain, and only one cofragment, the entire Z. If Z is aneven circuit and X 6=Z, then every cofragment of X on Z is an alternatingpath which starts and ends with an edge of Z−M . Thus, in this case, thefragments and the cofragments of X on Z are all alternating paths of oddlength (although cofragments are not chains).

The following theorem embodies the main idea of the proof of the MainTheorem.

Theorem 3.1. Let M = (G,F,M) be a bridgeless 2-mesh. If M has twoindependent transversal chains, then G contains perfect matchings F1 andF2 such that F1∩F2∩F =∅.


Proof. Let X1 and X2 be independent transversal chains inM. In order todescribe the 1-factors F1 and F2 fix i∈1,2, pick the chain Xi, and proceedas follows.

• Let Z be an arbitrary even circuit of M intersected by Xi. From eachfragment of Xi on Z include in Fi the edges occurring on odd positions,and from each cofragment of Xi on Z include in Fi the edges occurringon even positions. Repeat this process for every even circuit Z of Mintersected by Xi. As a result, Fi will include all F -edges of Xi, all non-M -edges of the fragments, and all M -edges of the cofragments of Xi onevery even circuit of M.• From every even circuit of M not intersected by Xi include in Fi all itsM -edges (or all its (Z−M)-edges – it does not matter).• Finally, add to Fi the maximum matching of A that misses both end-

vertices of Xi.

It is straightforward to verify that each Fi is a 1-factor with Fi ∩F ⊆Xi.Since the chains X1 and X2 are independent, we have X1∩X2∩F = ∅ andhence F1∩F2∩F =(F1∩F )∩(F2∩F )∩F ⊆X1∩X2∩F =∅. The result follows.

The difficult part of the proof of the Main Theorem, though, is to showthat the above construction can be applied to every bridgeless 2-mesh.

Theorem 3.2. Every bridgeless 2-mesh contains two independent transver-sal chains.

Although the proof of Theorem 3.2 occupies the rest of this paper, thereis one particular case where this theorem can be established immediately.Let us inspect the wires of an arbitrary mesh M= (G,F,M). Clearly, eachvertex of the odd part A of the 2-factor belongs to a unique wire, and exactlytwo vertices of A belong to the same wire. Since each circuit Z in A is odd,Z must be incident with an odd number of transversal wires. In particular, a2-mesh contains at least one transversal wire. If it contains two (and hence atleast three), these are the two independent transversal chains sought. In therest of the proof we can therefore assume that the mesh under considerationhas a single transversal wire. We call such a mesh singular and all other2-meshes non-singular. Furthermore, we call a 2-mesh good if it has twoindependent transversal chains, and call it bad otherwise.

Summing up, we have proved the following.

Proposition 3.3. Every non-singular 2-mesh is good.

The proof of the statement that every singular 2-mesh is good has severalstages. In the next section we introduce a reduction operation which from


a given bridgeless 2-mesh produces a “simpler” mesh, which has fewer evencircuits. By repeating this operation as many times as possible we arrive atso-called primitive meshes. Since the reduction of a bad 2-mesh is again bad(Proposition 4.1), the proof of Theorem 3.2 can be restricted to primitivemeshes. A particular advantage of using primitive meshes is that a primitivemesh can be endowed with a natural linear ordering of its circuits such thatA1 and A2 are the smallest and the largest element, respectively, and everyeven circuit is “avoided” by at most one ray leading from a predecessorcircuit to a successor circuit (Proposition 4.3). The latter property offers aneffective control over wires in a mesh and makes the ordering one of the maintools of our proof. For example, we can show that every transversal wire in aprimitive 2-mesh is “almost” increasing, that is to say, it may return to theimmediate predecessor of a circuit but not further back (Proposition 5.1).

A closer inspection of primitive meshes reveals that, in general, they neednot contain two independent transversal wires. This fact forces us to studymore general chains in a greater detail. In Section 6 we develop a machineryfor constructing new chains from old based on the idea of “changing” fromone chain to another. The resulting “transfer” technique is to be widelyapplied throughout the remainder of the proof. An immediate application oftransfer yields that a primitive 2-mesh having an increasing transversal wireis good (Proposition 6.2). This leaves us with singular primitive 2-mesheswhere the transversal wire returns at least once.

At this point it is natural to look at the behaviour of a transversalwire between consecutive circuits of a primitive 2-mesh. It transpires that atransversal wire may be moving back and forth between two consecutive cir-cuits arbitrarily many times, each time producing a “fold”. These folds forma barrier for a second transversal chain. In Section 7 we therefore replace ahighly folded transversal wire with a “smoother” alternating walk which lo-cally behaves like a chain and has only simple folds (Proposition 7.1). Eachfold of the resulting “quasichain” gives rise to three significant rays resultingfrom the changes of direction; we call this triple of rays a nest.

Section 8 starts with an investigation of chains connecting consecutivenests. We prove that two consecutive nests are connected by either two inde-pendent chains, both independent of the transversal quasichain, or by fourchains joining two pairs of rays such that each of the chains is independentof the quasichain (Proposition 8.1). With the help of this result we constructa new graph Υ , called a transfer graph for the mesh, whose vertex-set is theunion of all nests plus two additional vertices corresponding to the two oddcircuits, and whose edges correspond to specific chains between consecutivenests. Next we observe that certain pairs of internally disjoint increasing


transversal paths in Υ , provided that they exist, give rise to pairs of inde-pendent transversal chains in the mesh. In fact, these chains are obtained byglueing together chains that correspond to the edges of Υ (Proposition 8.4).Finally, we prove that such a pair of internally disjoint increasing transver-sal paths in Υ indeed exists (Proposition 8.6). This result eventually impliesthat every singular primitive 2-mesh with folded transversal wire is goodand concludes the proof of the Main Theorem.

4. Reduction and ordering

The first substantial step in the proof of Theorem 3.2 is a reduction to aclass of meshes with a more tractable structure. The reduction is based onan operation that takes an arbitrary meshM=(G,F,M) and an even circuitZ ofM and produces a smaller meshM′=(G′,F ′,M ′) in which the circuitZ is “dissolved” but all other circuits are retained as indicated in Fig. 1. Wenow describe this operation in detail.

To form the supporting graph G′ ofM′, we delete all the edges of Z−Mand suppress the resultant 2-valent vertices. The edges of G with no end-vertex in Z remain unaffected and G′ inherits them without any change. Theedges of G′ not directly inherited from G are in a one-to-one correspondencewith the fragments of wires of M intersecting Z. It follows that each edgee′ of G′ has a well-defined preimage σ(e′) in G: if e′ is not directly inheritedfrom G, then σ(e′) is the corresponding wire fragment, otherwise σ(e′)=e′.Note that the just described operation may produce a graph with paralleledges, bridges, and even separate connected components.

Figure 1. Dissolving a circuit Z


To construct the mesh M′ on G′, we let F ′ consist of all edges e′ of G′

such that σ(e′) is a chain in M. Thus F ′ includes all the F -edges that aredirectly inherited into G′ and all the edges of G′ that arise from fragmentsof wires intersecting Z. It is easy to see that F ′ is indeed a 1-factor of G′

and that its complementary 2-factor is C ′=G′−F ′=C−Z. Furthermore,the preimages of any two distinct edges e′ and f ′ from F ′ are independentchains σ(e′) and σ(f ′) inM. Finally, we let the cofactor M ′ ofM′ consist ofall edges of M that have been inherited into G′; in other words, M ′=M−Z.It is clear that M ′ is a 1-factor of the even part B′ = B−Z of C ′; henceM′=(G′,F ′,M ′) is a mesh.

Now let us assume thatM′ arises by the above operation from a 2-mesh.Then the two odd circuits of M fall into the same component N of M′, allother components of M′, if any, having only even circuits. Since the abovedescribed procedure creates no new odd circuits, N is again a 2-mesh. Wecall N a reduction ofM and say that N was obtained fromM by dissolvingthe circuit Z.

Let us call a 2-mesh M primitive if it is bridgeless and every reductionofM produces a mesh with a bridge. For example, a bridgeless 2-mesh withno even circuits is primitive. Our next proposition shows that in provingTheorem 3.2 it is enough consider primitive meshes.

Proposition 4.1. Let M be a 2-mesh. If a reduction of M is good, thenM is good.

Proof. LetM′=(G′,F ′,M ′) be a reduction ofM resulting from dissolvingan even circuit Z. Assume thatM′ is good. ThenM′ contains two indepen-dent transversal chains X ′1 and X ′2. If both X ′1 and X ′2 contain only edgesdirectly inherited from M, then X ′1 and X ′2 are independent transversalchains in M; hence M is good.

Assume that there exists an edge e′ in some X ′i that is not directlyinherited fromM. Then e′ is an F ′-edge and its preimage inM is the chainσ(e′). As mentioned above, the preimages of two distinct edges e′ and f ′

from F ′ are independent chains σ(e′) and σ(f ′) in M. If we substitute allsuch edges in X ′1∪X ′2 by the corresponding chains in M we produce a pairof transversal chains X1 and X2 in M. Since X ′1 and X ′2 were independent,so will be X1 and X2. Hence M is good, as claimed.

In dealing with primitive meshes the following lemma will often be useful.

Lemma 4.2. Let M be a 2-mesh. Then

(a) every edge-cut separating the two odd circuits from each other has oddsize, and vice versa;


(b) if M is bridgeless, it contains three independent transversal paths.

Proof. The statement (a) follows from a simple parity argument, while (b)is a direct consequence of Menger’s Theorem.

The next result describes a fundamental property of primitive 2-mesheswhich is best formulated in terms of an ordering of the set of circuits. Asimilar but weaker idea was used by Huck and Kochol in [4] to construct 5-cycle double covers in cubic graphs of oddness two. Before stating the resultwe need to develop some pertinent terminology and notation. Let “” bea linear ordering on the set of circuits of a mesh M. For a circuit Z ofM let Z− and Z+ denote, respectively, the immediate predecessor and theimmediate successor of Z in the ordering of M, provided that they exist.In addition, we set K− = K for the least element K of the ordering, andL+=L for the greatest element L of the ordering. We say that a ray e avoidsa circuit Z if it joins a circuit Z ′≺Z to a circuit Z ′′Z. Similarly, a walkavoids Z if it initiates in a circuit Z ′ ≺Z, terminates in a circuit Z ′′ Z,but does not intersect Z. If a walk avoids Z, then it necessarily contains aray that avoids Z.

Proposition 4.3. LetM be a primitive 2-mesh. Then there exists a uniquelinear ordering “” on the set of circuits of M such that

(O1) A1 and A2 are the smallest and the largest element, respectively,(O2) each circuit Z≺A2 is joined to its successor Z+ by some edge, and(O3) each circuit of M is avoided by at most one edge.

Furthermore, this ordering is independent of the cofactor of M.

Proof. Let M=(G,F,M) be a primitive 2-mesh with 2-factor C. To showthat the set of circuits of M admits a linear ordering satisfying (O1)-(O3)we examine the structure of the quotient graph G/C which arises from Gby contracting each circuit of C into a single vertex. Given a subgraph Lof G, we let L/C denote the image of L in the contraction G→G/C. Notethat the contraction may create multiple edges and loops even when G wassimple.

By Lemma 4.2,M contains three independent transversal paths. Hence,in the quotient graph G/C there are three edge-disjoint paths connectingthe two vertices a1=A1/C and a2=A2/C of odd valency.

Claim 1. Any two edge-disjoint a1-a2-paths in G/C together cover all thevertices of G/C.

Indeed, if P1 and P2 are two edge-disjoint a1-a2-paths in G/C, then inM there exist independent transversal paths P1 and P2 (not necessarily


chains) such that P1/C = P1 and P2/C = P2. We claim that P1 and P2

together intersect all even circuits ofM. Suppose not, and let an even circuitZ be intersected by neither P1 nor P2. If we dissolve Z, we obtain a 2-mesh M′ in which P1 and P2 survive as two independent A1-A2-paths.Lemma 4.2 (a) implies that M′ is bridgeless contradicting the assumptionthat M is primitive. This proves Claim 1.

Claim 2. Let P,Q,R be a set of three edge-disjoint a1-a2-paths in G/Cwith minimum total length. If a pair of vertices u and v occurs on two orthree of the paths, then the paths traverse u and v in the same order.

If the claim is false, then there exist vertices u and v of G/C such that,say, P and Q can be expressed in the form P = P1P2P3 and Q=Q1Q2Q3

where P2 is a u-v-path and Q2 is a v-u path. However, P ′=P1Q3, Q′=Q1P3,

and R constitute three edge-disjoint a1-a2-paths with a smaller total length,contradicting the choice of P,Q,R. Claim 2 is proved.

Claim 3. For any two vertices c and c′ of G/C there is a path among P ,Q, and R that contains both c and c′.

By Claim 1, c and c′ occur together in each of P ∪Q, P ∪R, and Q∪R.If, say, c lies on P but not on Q while c′ lies on Q but not on P , then clies on R, because it is contained in Q∪R, and c′ lies on R because it iscontained in P ∪R. So R contains both c and c′. Similar arguments hold forany permutation of P , Q, and R.

Claim 4. The set P,Q,R determines a linear ordering “” on the set ofcircuits ofM satisfying (O1), (O2), and (O3). This ordering is independentof the cofactor of M.

By Claim 1, the subgraph P ∪Q∪R comprises all the vertices of G/Cand, by Claim 2, the vertices of G/C can be partially ordered in such a waythat each of the three paths traverses its vertices in an increasing order.Claim 3 further implies that any two vertices of G/C are comparable underthis partial ordering. Hence the ordering of the vertices of G/C is linear.We now order the set of circuits of M simply by setting Z Z ′ wheneverZ/C precedes Z ′/C. The latter ordering is clearly linear and fulfils (O1) and(O2). We prove that it also satisfies (O3).

Suppose, to the contrary, that a circuit Z ofM is avoided by two distinctedges e and f . By Claim 3, there exist paths Pe and Pf in P,Q,R thatcontain the end-vertices of e and f , respectively. If one of the edges, say e,does not belong to P ∪Q∪R, we replace the portion of Pe delimited by theend-vertices of e with e itself to obtain a triple P ′,Q′,R′ of edge-disjointa1-a2-paths with a smaller total length, contradicting the choice of P,Q,R.It follows that both e and f are included in P ∪Q∪R. Because each of the


paths traverses the vertices of G/C in an increasing order, Pe and Pf mustbe distinct. As both of them miss the vertex Z/C of G/C, we have arrivedat a contradiction with Claim 1. This verifies (O3).

Finally, the construction of the ordering does not involve the cofactor ofM, so the ordering is independent of it, as claimed.

Claim 5. There exists a unique linear ordering satisfying (O1)-(O3).Suppose that there exists a linear ordering “0” that satisfies (O1)-(O3)

and is different from the one “” constructed above. Then there exists acircuit ZA1 whose successor U with respect to “0” differs from Z+, thesuccessor with respect to the original ordering “”. Choose Z to be minimalwith respect to “”. Since both orderings agree up to Z, we have U Z+

and Z+0U . In view of (O2), there is an edge e connecting Z to Z+ and anedge f connecting Z to U . By Lemma 4.2 (a), there exists an edge g joininga circuit V Z+ to a circuit W Z. Now Z+ is avoided by f and (O3) holds,so Z+ cannot be avoided by g. Therefore V =Z+ and g joins Z+ to W . Theminimality of Z also implies that W 0Z, whence W 0Z≺0U≺0Z

+. Butthen, with respect to the ordering “0”, both e and g avoid U , violating(O3). The proof is now complete.

The linear ordering of the set of circuits of a primitive 2-mesh M estab-lished in Proposition 4.3 will be simply called the ordering of M.

5. Folds

The ordering of circuits whose existence and uniqueness we have demon-strated in the previous section will now be applied to a detailed study ofwires in a primitive 2-mesh. The main result of this section is a descriptionof a “travel log” of a transversal wire from its journey between the smallestelement A1 and the largest element A2. This result reveals an importantfeature of transversal wires, the existence of folds, which will be discussedlater in this section.

Before proceeding to the result, we need several definitions. Let M be aprimitive 2-mesh. Given two circuits Z ′ and Z ′′ ofM we can form the closedinterval subgraph [Z ′,Z ′′]G as the subgraph induced by the set of verticesof all circuits Z such that Z ′ Z Z ′′. In particular, if Z ′ =Z ′′ =Z, then[Z ′,Z ′′]G = [Z]G. We also define the open interval subgraph (Z ′,Z ′′)G bysetting (Z ′,Z ′′)G=[Z ′,Z ′′]G−([Z ′]G∪[Z ′′]G). Half-closed interval subgraphs[Z ′,Z ′′)G and (Z ′,Z ′′]G can be defined analogously.

Let X be a chain in M. Define the log of X to be a sequence `(X) ofcircuits of M such that each entry Z in `(X) corresponds to a fragment of


X on Z and the entries occur in the order as the fragments are traversed byX; if X is closed, we regard `(X) as being a cyclic sequence. The followingresult describes logs of wires.

Proposition 5.1. LetM=(G,F,M) be a primitive 2-mesh and let R be awire of M.

(i) If R is a transversal wire with log `(R)=(Z1,Z2, . . . ,Zm), then ZjZ−iwhenever j>i.

(ii) If R is a local wire, then `(R) contains at most two different circuits,which must be consecutive with respect to the ordering of M.

Proof. Let R be a transversal wire of M. Suppose to the contrary that Mhas circuits Y and Z with Y ≺Z− such that a traversal of R through [Z]Goccurs before a traversal through [Y ]G; in other words, an occurrence of Zin `(R) is followed by an occurrence of Y . Clearly, Z− is an even circuit.Our aim is to show that we can dissolve Z− to produce a bridgeless 2-mesh,contradicting the assumption that M is primitive (see Fig. 2).

Let M′ = (G′,F ′,M ′) be the reduction of M resulting from dissolvingZ−. Then the circuits of M′ are linearly ordered by means of the orderinherited from M. If there is a bridge in G′, then it must be an edge of F ′.We therefore examine edge-cuts in G′ entirely consisting of F ′-edges andshow that each such cut has at least two edges.

Claim 1. For each circuit U of M′ let H ′U be the cut separating the sub-graph [A1,U ]G′ from the rest of G′. Then |H ′U |≥3.

Let HU be the cut separating the subgraph [A1,U ]G from the rest of G.Then |HU | ≥ 3 by Lemma 4.2. To show that |H ′U | ≥ 3 we consider severalpossibilities depending on the position of U inM with respect to the circuitW =Z−−. If U≺W , then U≺W ≺Z−. By Proposition 4.3, W is avoided byat most one edge, therefore at most one edge of HU leads to Z−. DissolvingZ− then leaves at least two edges of HU intact, and these edges are inheritedinto H ′U . It follows that |H ′U | ≥ 2 and hence, by Lemma 4.2, |H ′U | ≥ 3. IfU Z, then Z− ≺ Z ≺ U+, and by similar arguments |H ′U | ≥ 3. Thus itremains to consider the cut H ′W . The assumption on `(R) now implies thatR can be written in the form R = R1R2R3 where R2 is a Z-Y -path. Bycondition (O3) of Proposition 4.3, at least two of the paths R1, R2, andR3 traverse Z−. As G′ arises from G by dissolving Z−, each of the twopaths contains a subchain that transforms into an edge of G′. The resultingedges clearly belong to H ′W , so |H ′W |≥ 2 and consequently, by Lemma 4.2,|H ′W |≥3. This establishes Claim 1.

Claim 2. Every even circuit of M′ is avoided by at most one edge.


Figure 2. Reduction in the proof of Proposition 5.1

Suppose that a circuit U of M′ is avoided by two distinct edges e′1 ande′2. Then each e′i belongs to F ′, connects a circuit U ′i to a circuit U ′′i suchthat U ′i≺U≺U ′′i , and its preimage in G is the chain σ(e′i) connecting U ′i toU ′′i and traversing Z−. Note that the chains σ(e′1) and σ(e′2) are independentbecause e′1 6=e′2, and neither of them traverses U . Furthermore, the inequalityU ′i ≺U ≺U ′′i holds also in M, so each σ(e′i) avoids U and hence it containsan edge fi ∈F avoiding U . As σ(e′1) and σ(e′2) are independent, f1 and f2are distinct. Thus U is avoided in M by two distinct edges, contradictingProposition 4.3. This proves Claim 2.

Now let J ′ be an arbitrary edge-cut of G′ contained in F ′; we show that|J ′|≥ 2. Let U be the first circuit of M′ with the property that U and U+

belong to different components of G′−J ′. Consider the cut H ′U separating[A1,U ]G′ from the rest of G′. By Claim 1, |H ′U |≥3, and by Claim 2, at mostone edge of H ′U avoids U+. Thus at least two edges of H ′U are contained inJ ′ implying that |J ′|≥2.

All the above proves that the reductionM′ ofM is bridgeless, contradict-ing the assumption that M is primitive. Summing up, if R is a transversalwire, its log `(R) has the desired property.


It remains to consider the case that R is a local wire. Then R is eitherclosed or has its end-vertices in the same odd circuit A1 or A2. Since everycircuit can be avoided by at most one edge of R, the set of circuits havingnon-empty intersection with R forms an interval. If this interval containsmore than two circuits, then any inner circuit within it can be dissolved toproduce a bridgeless reduction ofM, contradicting the assumption thatMis primitive. Details are similar to those above and therefore are left to thereader. The proof is complete.

Corollary 5.2. In a primitive 2-meshM, every edge that avoids some cir-cuit ofM is contained in a transversal wire, which traverses the edge in theincreasing direction.

Proof. Let e be an edge that avoids a circuit of M, and let T be the wirecontaining e. By Proposition 5.1 (ii), T must be transversal, and by Part (i)of the same proposition it must traverse e in the increasing direction.

Before drawing further consequences from Proposition 5.1 we need a fewmore definitions. Let X be chain and let Z be a circuit of a primitive 2-meshM. Define a fold of X at Z as an inclusion-minimal contiguous subsequenceof `(X) of the form (U,. . . ,V, . . . ,U ′) where both U Z and U ′ Z, butV Z. Recall that each occurrence of a circuit W in `(X) represents afragment of X on W . If we take the union of all fragments correspondingto the circuits from (U,. . . ,V, . . . ,U ′) strictly between U and U ′ we obtain asubchain of X that uniquely represents the fold. A short reflection furthershows that different folds give rise to disjoint subchains. The lower part ofFig. 2 illustrates this concept by showing a fold of a transversal wire R atcircuit Y .

The folding number of X at Z, denoted by φ(X,Z), is the number offolds of X at Z. A chain X is said to be smooth in the subgraph [Z,Z ′]G ifφ(X,U) = 0 for every circuit U such that ZU ≺Z ′; otherwise X is foldedin [Z,Z ′]G. If X is smooth in [A1,A2]G=G, we simply say that it is smooth;otherwise X is folded.

Consider a wire R in a primitive 2-meshM. What can we say about thebehaviour of R near a circuit Z? In particular, what is the folding numberφ(R,Z)? It follows immediately from the definitions that φ(R,A2) = 0 andφ(R,A1)≤1, the value 1 occurring only when R is a local wire incident withA1. Let Z be an even circuit. If R avoids Z+, or if R is stationary, then againφ(R,Z)=0. So assume that R reaches Z+ with some ray e whose other endlies in a circuit UZ. In [Z+]G, the wire R continues along the fragment ofR on Z+ containing e until it leaves [Z+]G with a ray f . By Proposition 5.1,the edge f either leads to a circuit Z ′ Z+, in which case φ(R,Z) = 0,


or turns back to the circuit Z, producing a fold at Z. This fold may befollowed by a number of other folds at Z, but all such folds are confinedwithin [Z,Z+]G. One of the difficulties that we have to face in the proofof Theorem 3.2 is that φ(R,Z) may be arbitrary large. In other words, Rmay “oscillate” between Z and Z+ any number of times before it ultimatelyproceeds to a circuit Z ′Z+. In fact, R may use up all the edges of the cutHZ separating Z from Z+; if this is the case, then M cannot contain twoindependent transversal wires. To summarise, in proving Theorem 3.2 onecannot get along with wires only and needs to consider pairs of transversalchains neither of which is a wire.

Figure 3. Simple transfer

6. Transfer

The aim of this section is to develop tools for effective construction of chainsfrom their basic building blocks – wires and even circuits. The principalidea is that a new chain can be created from two chains X1 and X2 alreadyconstructed by “changing” from X1 to X2 on an even circuit where both X1

and X2 meet.The following operation will often be used for this purpose. Let W be an

e-f -walk and let W ′ be an f -g-walk, both traversing f in the same direction.Then W and W ′ can be linked through this common edge into an e-g-walkW tW ′. Clearly, this operation is associative.

Consider an even circuit Z of a meshM and let e be a ray of Z. We maythink of e as an edge of our starting chain. Let P be a non-stationary chaininM which has an edge in common with Z; this is our target chain. In Z, wetake the shortest path O starting with the M -edge g adjacent to e and endingwith an edge in P ∩Z, say h. Then there is a unique chain of the form W =


eOP ε(h,∗], the choice of P ε ∈ P,P−1 being unambiguously determinedby the requirement that W be a chain. Denoting by f the first ray on Wfollowing e we can define the transfer function ω by setting ω(e,P ;Z)=f andthe transfer chain Ω(e,P ;Z) from e to P on Z to be the chain W [e,f ]. Thewhole operation is illustrated in Fig. 3. Note that the definition of a transfermakes sense even when e is contained in P : in this case the path O consistsof a single edge g=h, and consequently Ω(e,P ;Z) = egP ε(g,f ] =P ε[e,f ] isa fragment of P ε on Z. Thus the transfer function ω( · ,P ;Z) is a mappingfrom the set of rays of Z into itself such that for each ray e of Z the edgeω(e,P ;Z) is a ray contained in P .

The form of a transfer that we have just introduced can be called asimple transfer. Under certain conditions, simple transfers can be composedinto multiple transfers. To make this idea work we introduce the concept ofa tube between two circuits. Consider two even circuits Y Z in a primitive2-meshM, not excluding the case that Y =Z. An ascending tube from Y toZ is a sequence K=(KU ; Y UZ) of non-stationary chains KU , indexedby the circuits between Y and Z, such that each KU has a common edgewith U , and for U≺Z the chain KU is contained in [U,U+]G. This definitionis illustrated in the upper part of Fig. 4.

Figure 4. Transfer through an ascending tube

Assume that there exists an ascending tube K= (KU ; Y U Z) fromY to Z. If we take an arbitrary ray e of Y , then, by the definition of asimple transfer, we can calculate the ray eY =ω(e,KY ;Y ) and the transfer


chain SY = Ω(e,KY ;Y ). Since KY is contained in the induced subgraph[Y,Y +]G, the ray eY joins Y to Y +. If ZY , we may calculate the ray eY + =ω(eY ,KY + ;Y +), and so on. It follows that there exists a unique sequence(eU ; Y UZ) of rays such that

• eY =ω(e,KY ;Y ),• eU =ω(eU− ,KU ;U) for each U with Y ≺UZ, and• eU joins U to U+ for each U with Y U≺Z.

Along with this sequence, we also have the corresponding sequence of trans-fer chains (SU ; Y UZ) where SY =Ω(e,KY ;Y ) and SU =Ω(eU− ,KU ;U)for each UY . The last edge eU of each SU coincides with the first edge ofSU+ , whenever defined, and this common edge is traversed by both chains inthe same direction. Hence we can construct the walk S=

⊔(SU ; Y UZ).

It is immediate that S is alternating and its outer edges e and eZ are rays.Furthermore, each SU (eU− ,eU ) is a path contained in the induced subgraph[U ]G, therefore S does not repeat edges. It follows that S is a chain.

These considerations enable us to extend the definitions of the transferfunction and the transfer chain to ascending tubes as follows. We definethe transfer function through the ascending tube K by setting ω(e,K)=eZand the transfer chain through K by setting Ω(e,K) = S. We stress thatboth ω(e,K) = eZ and Ω(e,K) are uniquely determined by the ray e andthe tube K; in particular, ω( · ,K) is a mapping from the set of rays of Yto the set of rays of Z. This suggests that we may think of an ascendingtube K as a device which receives a ray e at its beginning, the circuit Y ,and returns a well-defined ray f=ω(e,K) at its end, the circuit Z, togetherwith a uniquely defined e-f -chain Ω(e,K) (see the lower part of Fig. 4). Auseful property of the resulting chain Ω(e,K) is that each F -edge of Ω(e,K)different from e belongs to one of the chains KU constituting K, that is tosay, Ω(e,K)∩F ⊆e∪

⋃UKU .

Transfers may also be performed in the reverse direction. For this purposewe introduce the concept of a descending tube. Let Y Z be two circuitsof a primitive 2-mesh M. A descending tube from Z to Y is a sequenceL= (LV ; Z V Y ) of non-stationary chains LV such that each LV has acommon edge with V , and for V Y one has LV ⊆ [V −,V ]G. Note that adescending tube from Z to Y is nothing but an ascending tube from Z toY with respect to the reverse ordering of M.

Given a descending tube L= (LV ; ZV Y ) and an arbitrary ray g ofZ, we again calculate the ray gZ =ω(g,LZ ;Z). If Y ≺Z, the resulting ray gZjoins Z to Z−, because the chain LZ is contained in the induced subgraph[Z−,Z]G. Thus there exists a unique sequence (gV ; Z V Y ) of rays gVsuch that


• gZ =ω(g,LZ ;Z)• gV =ω(gV + ,LV ;V ) for each V with ZV Y , and• gV joins V to V − for each V with ZV Y .

Together with the sequence (gV ; Z V Y ) we have the correspondingsequence of transfer chains (TV ; ZV Y ) where TZ =Ω(g,L;Z) and TV =Ω(gV + ,LV ;V ) for V ≺Z. Any two consecutive chains TV and TV − traversetheir common edge gV in the same direction, so we can link the chains into ag-gY -chain T =

⊔(TV ; ZV Y ). As above, the ray gY and the chain T are

uniquely determined by g and the tube L. We therefore define ω(g,L)=gYand Ω(g,L)=T and call them the transfer function and the transfer chainthrough the descending tube L from Z to Y , respectively. Observe that forthe transfer chain we similarly have Ω(g,L)∩F ⊆g∪

⋃V LV .

An ascending and a descending tube may sometimes together form adouble tube. Again, let Y Z be two circuits of a primitive 2-mesh M. Adouble tube between Y and Z is a pair (K,L) consisting of an ascending tubeK = (KU ; Y U ≺Z) from Y to Z− and a descending tube L = (LV ; Z V Y ) from Z to Y + such that

• KU =LU+ for each circuit U with Y U≺Z, and• KU ∩LU =∅ for each circuit U with Y ≺U≺Z.

If Y =Z, then K=L=∅, producing the empty double tube (∅,∅).Observe that in a double tube both tubes have the same set of chains,

although playing different transfer roles. Hence, a double tube between twocircuits Y and Z with Y Z can be alternatively described as a set of non-stationary chains D = XU ; Y U ≺ Z such that (1) each chain XU iscontained in [U,U+]G and has a common edge with both U and U+, and(2) XU− ∩XU =∅ for Y ≺U≺Z.

A basic example of a double tube consists of a single local wireW betweena circuit Y and its successor Y +. Formally, D=(K,L) where K=(KY ) andL = (LY +) with KY =W = LY + . This example immediately generalises toa double tube D between two circuits Y and Z Y + which consists ofnon-stationary local wires filling up the interval [Y,Z]G, provided that therequired local wires exist. Our next lemma shows that if a transversal wireR is smooth in [Y,Z]G, then D indeed exists, and is independent of R in thesense of the following definition. An ascending or a descending tube L=(LU )is said to be independent of a chain X if LU ∩X ∩F = ∅ for each index U .A double tube (K,L) is independent of X if both K and L are independentof X.


Proposition 6.1. Let M be a singular primitive 2-mesh with transversalwire R, and let Y Z be two even circuits of M. If R is smooth in [Y,Z]G,then there exists a double tube between Y and Z independent of R.

Proof. If Y =Z, then the required double tube is the empty double tube(∅,∅). Assume that Y ≺ Z. Consider an arbitrary circuit of U of M suchthat Y U ≺ Z and take the edge-cut HU ⊆ F separating the subgraph[A1,U ]G from the rest of G. Lemma 4.2 shows that |HU | ≥ 3. Since R issmooth in [Y,Z]G, at most one edge of HU belongs to R. Furthermore, byProposition 4.3, U is avoided by at most one edge of HU . It follows thatHU −R includes an edge e connecting U to a circuit W U+. Let KU bethe wire containing e. Since KU is local, Proposition 5.1 implies that thelog of KU contains only two consecutive circuits. Thus `(KU ) = (U,U+); inparticular, KU has an edge in common only with U and U+, and with noother circuit of M. It follows that K = (KU ; Y U Z−) is an ascendingtube. By rewriting each KU as LU+ , the sequence L = (LV ; Z V Y +)becomes a descending tube.

To see that the pair (K,L) forms a double tube we only need to verifythat KU ∩LU =∅ for each circuit U with Y ≺U≺Z. Observe, however, thatLU =KU− for each such U . Hence `(LU ) = (U−,U) while `(KU ) = (U,U+).Therefore LU and KU are distinct wires, and so they are disjoint. Thus(K,L) is a double tube.

Finally, we verify that (K,L) is independent of R. Recall that each KU

is a local wire while R is a transversal wire. Therefore KU ∩R=∅, implyingthat (K,L) is independent of R.

Now we are ready for a further step forward in the proof of Theorem 3.2.

Proposition 6.2. Every primitive 2-mesh containing a smooth transversalwire is good.

Proof. Let M be a primitive 2-mesh and let R be a smooth transversalwire in M. Without loss of generality we may assume that M is singular.This particularly means that M has at least one even circuit. Consider thecircuits A+

1 and A−2 which are both even and A+1 A

−2 . Since R is smooth,

Proposition 6.1 guarantees that M has a double tube (K,L) between A+1

and A−2 independent of R. Further, sinceM is singular and bridgeless, thereexists a ray independent of R that joins A−2 to A2. Let L be the wire contain-ing this ray. Then L is local and therefore, by Proposition 5.1, it is containedin the subgraph [A−2 ,A2]G. Moreover, L has an edge in common with A−2 .Thus we can extend K to an ascending tube K′ from A+

1 to A−2 by adding Lto K as the last chain KA−2

. To finish the proof, consider an arbitrary chain


Ω(x,K′) where x is a ray of A1 independent of R. Since K′ is independentof R and Ω(x,K′)∩F ⊆x∪

⋃(KU ;A+

1 UA−2 ), it follows that Ω(x,K′),

too, is independent of R. Thus R and Ω(x,K′) form a pair of transversalchains, which makes M good.

7. Smoothing

As pointed out before the end of Section 5, the presence of folds in a transver-sal wire forces us to seek a pair of independent transversal chains outsidethe set of wires. In doing so, we shall next prove that every transversal wirein a primitive 2-mesh can be transformed into a transversal alternating walkwith properties similar to a chain and folding numbers at most 1. The re-sulting “quasichain” has the advantage that it offers a better control overthe transfer function, a feature that will be helpful in the next section.

We start with a series of important but rather technical definitions. LetX be a transversal alternating walk in a primitive 2-mesh M. With eachcircuit Z≺A2 of M we associate two significant rays fXZ and lXZ from X asfollows. For Z=A1 we define fXA1

= lXA1to be the first edge of X. Otherwise,

if Z is an even circuit, we let

fXZ be the first ray in X incident with a circuit UZ, and

lXZ be the last ray in X incident with a circuit UZ.

If the context is clear, the superscript X or the subscript Z can be omitted.Before proceeding to the next definition recall that a chain is an alter-

nating walk with no repeated edges which either begins and ends with a ray,or is closed. We now relax the restriction on repetitions by allowing repeti-tions of the edges of B but additionally we require certain segments to bechains. Let M be a primitive 2-mesh. Define a transversal quasichain to bea transversal alternating walk X in M which does not repeat F -edges andsatisfies the following two conditions for every even circuit Z of M:

(Q1) the closed segments X[fZ− , lZ− ], X[lZ− ,fZ ], and X[fZ , lZ ] are chainsin M, and

(Q2) X(fZ− , lZ−)⊆ [Z−,Z]G, X(lZ− ,fZ)⊆ [Z]G, and X(fZ , lZ)⊆ [Z,Z+]G.

In particular, in a primitive 2-mesh each transversal chain is a transversalquasichain.

Transversal quasichains often behave similarly as chains. For example,one can define a fragment of a transversal quasichain analogously as wehave defined a fragment of a chain in Section 3. We also define the folding


number of a transversal quasichain X at a circuit Z as the folding numberof the chain X[fXZ , l

XZ ] at Z; that is, we set φ(X,Z)=φ(X[fXZ , l

XZ ],Z).

Given a transversal wire R we want to construct a transversal quasichainS with folding numbers at most 1 which coincides with R on all smoothsegments; we call such a quasichain a truncation of R. To be more precise,a transversal quasichain S is a truncation of R if it satisfies the followingthree conditions:

(S1) φ(S,Z)=minφ(R,Z),1 for each circuit Z of M,(S2) S[lU ,fV ]=R[lU ,fV ] for any two circuits U ≺V such that R is smooth

in [U+,V ]G, and(S3) S∩F ⊆R.

We now prove the following.

Proposition 7.1. Every transversal wire in a primitive 2-mesh has a trun-cation.

Proof. Let M be a primitive 2-mesh and let R be a transversal chainin M. If φ(R,U) ≤ 1 for each circuit U of M, then R itself satisfies thestatement. Otherwise there exist a circuit Z such that φ(R,Z) > 1. Wedescribe a procedure which takes a transversal quasichain T and a circuit Zwith φ(T,Z)>1 and produces a new transversal quasichain T ′ such that

• φ(T,Z)>φ(T ′,Z)≥1,• φ(T ′,U)=φ(T,U) for each circuit U of M other than Z, and• T ′∩F ⊆T .

When this procedure is applied to the wire R and repeated as long as nec-essary for all circuits Z with φ(R,Z) > 0 the eventual result will be therequired truncation S.

Consider a transversal quasichain T such that φ(T,Z) = α > 1 forsome circuit Z of M. For the rest of the proof Z remains fixed. Takethe segment J = T [fTZ , l

TZ ] of T ; by (Q1), J is a chain. It follows from

the definition of the folding number that the rays in J form a sequencefTZ =x1,y1,x2,y2, . . . ,xα,yα,xα+1 = lTZ such that each J [xi,yi] is a fragmentof J on Z+ and each J [yi,xi+1] is a fragment of J on Z. We distinguish twocases according to whether ω(x1,J [x2,∗];Z+)=yr or ω(x1,J [x2,∗];Z+)=xrfor some r ≥ 2. The latter case will be subdivided into two subcases de-pending on whether ω(xα+1,J [∗,xr];Z) = ys, where 1 ≤ s ≤ r − 1, orω(xα+1,J [∗,xr];Z)=xs, where 2≤s≤r.

Case 1. Assume that ω(x1,J [x2,∗];Z+)=yr for some r≥2. In this case,form T ′ from T by replacing the subchain T [x1,yr] of J with the transferchain Ω(x1,J [x2,∗];Z+). This transformation is indicated in Fig. 5.


Figure 5. Case 1 of Proposition 7.1

Clearly, T ′ is a transversal walk. We wish to show that T ′ is a transver-sal quasichain and has all the above stated properties. First observe thatthe fragments of T ′ and T on each circuit U other than Z and Z+

are the same. As regards Z and Z+, the fragments of T ′ on Z are thechains J [yr,xr+1],J [yr+1,xr+2], . . . ,J [yα,xα+1] while the fragments of T ′ onZ+ are the chains Ω(x1,J [x2,∗];Z+) and J [xr+1,yr+1],J [xr+2,yr+2], . . . ,J [xα,yα]. It follows that T ′ is a transversal alternating walk containedin F ∪B. From the construction of T ′ it is clear that for each even cir-cuit U of M we have fT

′U = fTU and lT

′U = lTU . Note that T and T ′

can only differ within the open segment T (fZ , lZ). To verify (Q1) for T ′

note that T ′[fZ− , lZ− ] = T [fZ− , lZ− ] and T ′[lZ− ,fZ ] = T [lZ− ,fZ ]. Further-more, T ′[fZ , lZ ] =Ω(x1,J [x2,∗];Z+)T (yr, lZ ], which is clearly a chain. Be-cause T satisfies (Q2) for every even circuit, so does T ′ except possiblyfor Z and Z+. However, T ′(fZ− , lZ−) ⊆ [Z−,Z]G, T ′(lZ− ,fZ) ⊆ [Z]G, andsince T ′[fZ , lZ ] ⊆ T [fZ , lZ ]∪ [Z+]G, we also have T ′(fZ , lZ) ⊆ [Z,Z+]G. AsT ′[fZ , lZ ] ⊆ T [fZ , lZ ]∪ [Z+]G, we have T ′(fZ , lZ) ⊆ [Z,Z+]G, T ′(lZ ,fZ+) ⊆[Z+]G, and T ′(fZ+ , lZ+)⊆ [Z+,Z++]G. Thus T ′ satisfies (Q2) for every evencircuit, which in turn implies that T ′ is a transversal quasichain.

It is easy to check that 1≤φ(T ′,Z)=φ(T,Z)−r+1=α−r+1<φ(T,Z)and φ(T ′,U) =φ(T,U) for each circuit U 6=Z. Since T ′[∗,fZ ] =T [∗,fZ ] andT ′[lZ ,∗] =T ′[lZ ,∗], the quasichain T ′ satisfies (S2). Finally, it is immediatefrom the construction that each F -edge in T ′ has already been used in T ,verifying (S3). This concludes Case 1.

Case 2.1. Assume that ω(x1,J [x2,∗];Z+) = xr for some r ≥ 2. Ifω(xα+1,J [∗,xr];Z) = ys for some s such that 1 ≤ s ≤ r − 1, thenwe form T ′ by replacing the subchain T [ys,xα+1] of J with the chain(Ω(xα+1,J [∗,xr];Z))−1 as indicated in Fig. 6. In a similar way as above one


Figure 6. Case 2.1 of Proposition 7.1

can check that T ′ is an alternating walk satisfying all the properties thatmake it a transversal quasichain. Moreover, 1≤φ(T ′,Z)=s<r≤α=φ(T,Z)and φ(T ′,U) =φ(T,U) for each circuit U 6=Z. As above, (S2) and (S3) areclearly fulfilled.

Figure 7. Case 2.2 of Proposition 7.1

Case 2.2. Finally assume that ω(x1,J [x2,∗];Z+) = xr for somer ≥ 2 and that ω(xα+1,J [∗,xr];Z) = xs for some s such that 2 ≤s ≤ r. We form T ′ by replacing the segment J of T with the chainΩ(x1,J [x2,∗];Z+)(T (xs,xr))

−1(Ω(xα+1,J [∗,xr];Z))−1. This transformationis illustrated in Fig. 7. Again, one can easily check that T ′ is an alternatingwalk satisfying all the properties that make it a transversal quasichain. Fur-thermore, 1≤φ(T ′,Z)=r−s+1≤α−s+1<α=φ(T,Z) and φ(T ′,U)=φ(T,U)for U 6=Z. Also (S2) and (S3) are clearly fulfilled. This concludes the Case2.2, and the proof follows.


8. Synthesis

Propositions 3.3, 4.1, and 6.2 have reduced the proof of Theorem 3.2 tosingular primitive 2-meshes with folded transversal wire. The purpose ofthis section is to resolve the remaining case by combining the ideas fromprevious sections.

Throughout the whole sectionM will denote a singular primitive 2-meshwith folded transversal wire and R will denote the wire. Let Φ be the set ofall circuits D of M such that either φ(R,D)>0 or D is odd. We view Φ asan ordered set with respect to the ordering inherited from M and denoteby D1 and D2 the smallest and the largest even circuit in Φ. For a circuit Din Φ we let D] and D[ denote the immediate successor and the immediatepredecessor of D in Φ, respectively, whenever they are defined. In particular,(A1)

]=D1 and (A2)[=D2.

Let S be a truncation of R. With each circuit D ∈ Φ we now associatethree symbols pD, qD, and rD, and form the set ΘD = pD, qD, rD calledthe nest of D. First, let D be even; thus φ(R,D)> 0 and φ(S,D) = 1. Weset pD=fSD, rD= lSD, and define qD to be the first ray in S following pD, infact, the only ray between pD and rD. If D=Ai, where i=1,2, we form thenest ΘD = ai by setting ai = pAi = qAi = rAi =Ai. With these definitions,S[pD, qD] is a fragment of S on D+ and S[qD, rD] is a fragment of S on D.Furthermore, by condition (S2), S[rD,pD] ] = R[rD,pD] ], so S[rD,pD] ] is achain.

Observe that the set Φ is uniquely determined by M whereas the nestsdepend on the choice of S. As the choice is irrelevant, we will not indicateit in our notation.

We now investigate chains in M that connect consecutive nests.

Proposition 8.1. Let M be a singular primitive 2-mesh with foldedtransversal wire R, and let S be a truncation of R. Then, for every circuitD≺A2 in Φ, at least one of the following holds.

(a) The subgraph (D,D]+)G contains two independent pD, qD-qD] , rD]-chains, both independent of S[rD,pD] ].

(b) The subgraph (D,D]+)G contains an x-y-chain for each x ∈ pD, qDand each y∈qD] , rD], all independent of S[rD,pD] ].

Proof. Consider an arbitrary circuit D of Φ such that D≺A2. We distin-guish two cases according to the position of D.

Case 1. Assume that A1≺D≺D]≺A2. Since R is smooth in [D+,D]]G,by Proposition 6.1 we can choose a double tube (K,L) between D+ and D]

independent of R. We extend the ascending tube K=(KU ;D+U≺D]) to


an ascending tube K′=(KU ;D+UD]) by setting KD] =S[qD] , rD] ]. Wealso extend the descending tube L=(LV ;D]V D+) to a descending tubeL′ = (LV ;D] V D+) by setting LD+ = S[pD, qD]. Both extensions arecorrectly defined: by the definition of a transversal quasichain, S[qD] , rD] ]and S[pD, qD] are non-stationary chains, and by the definition of Φ, theyhave an edge in common with D] and D+, respectively.

Next we make two observations.

Claim 1. For each x∈pD, qD and each y∈qD] , rD] we have ω(x,K′)∈qD] , rD] and ω(y,L′)∈pD, qD.

Take x∈pD, qD and consider the element x′=ω(x,K). It is easy to seethat x′ is a ray connecting D]− to D]. Now ω(x,K′)=ω(x′,S[qD] , rD] ];D]),so we have to show the latter to be one of qD] and rD] . Observe thatS[qD] , rD] ] is a fragment of S on D]. In other words, the only rays inS[qD] , rD] ] are qD] and rD] . Therefore ω(x′,S[qD] , rD] ];D]) = ω(x,K′) ∈qD] , rD], as claimed. The proof that ω(y,L′)∈pD, qD is similar.

Claim 2. For each x∈pD, qD and each y∈qD] , rD], the chains Ω(x,K′)and Ω(y,L′) are independent of S[rD,pD] ].

To see that the claim is true, take an arbitrary edge x ∈ pD, qD andrecall that from the definition of a transfer chain we have

Ω(x,K′) ∩ F ⊆ x ∪⋃KU ; D+ U D].

Since the chains KU , where U ≺D], are all independent of R and hence ofS, we see that

S[rD, pD] ] ∩Ω(x,K′) ∩ F ⊆ S[rD, pD] ] ∩ (x ∪ S[qD] , rD] ]) ∩ F = ∅.

Thus Ω(x,K′) is independent of S[rD,pD] ]. Similarly, each Ω(y,L′) is inde-pendent of S[rD,pD] ]. This proves the claim.

Consider an arbitrary edge x∈pD, qD. In qD] , rD], pick the elementy distinct from ω(x,K′) and denote the edge ω(y,L′) by τ(x). Note thatτ(x)∈pD, qD is uniquely determined by x; thus τ is a mapping of pD, qDinto itself.

Subcase 1.1. Assume that there exists x∈pD, qD such that τ(x) 6=x.As above, let y be the element of qD] , rD] such that ω(y,L′) = τ(x). SetX = Ω(x,K′) and Y = Ω(y,L′). If X and Y are independent, then (a)holds. Otherwise X and Y have an F -edge e in common. We claim that ecannot be a chord. Suppose that e is a chord of some circuit Z of M. ThenD+ Z D] and e ∈X ∩Y ∩ [Z]G ∩F . Recall that X is a transfer chainthrough an ascending tube while Y is a transfer chain through a descending


tube. Therefore X∩[Z]G⊆Z∪KZ and Y ∩[Z]G⊆Z∪LZ . Furthermore, since(K,L) is a double tube, we have KZ ∩LZ =∅ and hence

e ∈ X ∩ Y ∩ [Z]G ∩ F ⊆ (Z ∪KZ) ∩ (Z ∪ LZ) ∩ F = ∅,

which is absurd. Therefore e is a ray contained in a certain chain KW fromthe tube K, so e joins the circuit W to its successor W+. Now X can bewritten as X1eX2 where both X1e and eX2 are chains; similarly Y can bewritten as Y1e

−1Y2 where both Y1e−1 and e−1Y2 are chains. It follows that

X, Y −1, X1eY−11 , and Y −12 eX2 are alternating walks joining each edge in

pD, qD with each edge in qD] , rD]. Since both X and Y are indepen-dent of S[rD,pD] ], so are the walks X1eY

−11 and Y −12 eX2. To see that they

are actually chains, it is sufficient to realise that X1 and Y2 are containedin (D,W ]G, while X2 and Y1 are contained in [W+,D]+). Hence none ofthem can have repeated edges. Summing up, if X and Y have an F -edge incommon, then X, Y −1, X1eY

−11 , and Y −12 eX2 are chains that establish (b).

Subcase 1.2. Assume that τ(x)=x for each x∈pD, qD. Take x=pD,set u= ω(pD,K

′), and let w be the other element of qD] , rD]. Then, byour assumption, ω(w,L′) = pD. Consider the element ω(qD,K

′) ∈ u,w.If we had ω(qD,K

′) = u, we would get τ(qD) 6= qD, contrary to the as-sumption. Therefore ω(qD,K

′)=w. Similarly, ω(u,L′) must be distinct frompD, whence ω(u,L′) = qD. It follows that the chains Ω(pD,K

′), Ω(qD,K′),

Ω(qD] ,L′)−1, and Ω(rD] ,L′)−1 fulfil (b). This completes Case 1.

Case 2. Assume that either D=A1 or D]=A2. We only consider the firstpossibility, because the second one is analogous. By Proposition 6.1, thereexists a double tube (K,L) between A+

1 and D1. Select any ray e of A1 inde-pendent of S, and let L be the local wire containing e. By Proposition 5.1,L has an edge in common with A+

1 and with no other circuit of M. Thuswe can extend the descending tube L= (LV ;D1V A+

1 ) to a descendingtube L′ = (LV ;D1 V A+

1 ) by setting LA+1

= L. Then Ω(qD1 ,L′)−1 and

Ω(rD1 ,L′)−1 are easily seen to be the required chains satisfying (b). This

completes the proof.

From now on, we fix a truncation S of the transversal wire R in a sin-gular primitive 2-mesh M. With the help of Proposition 8.1 we constructa simple graph Υ , called a transfer graph for M, whose vertex-set will bethe union of all nests and whose edges will correspond to certain chainsbetween consecutive nests (see Fig. 8). The transfer graph Υ will serve asa device for identifying two independent transversal chains in M. In Υ weconstruct a pair of two increasing internally disjoint a1-a2-paths and show


that they correspond to two independent transversal chains in M, therebyestablishing Theorem 3.2.

Figure 8. Constructing a transfer graph Υ

To describe the edge-set of Υ we take an arbitrary circuit D ∈ Φ withD≺A2 and list all the edges in the subgraph ΥD induced by the set ΘD∪ΘD]

whose elements are now viewed as vertices of Υ . Each edge of ΥD will haveone end in ΘD and the other end in ΘD] , so there will be no edges insidenests. We then set Υ =

⋃D≺A2

ΥD. With each edge st of Υ , where s ∈ΘDand t∈ΘD] for some D, we associate one particular s-t-chain Π(st) in Mcontained in the subgraph (D,D]+)G and say that the edge st corresponds tothe chain Π(st), and vice versa. The edges of ΥD and the chains are definedas follows.

(T1) First of all, for each D≺A2 we include in ΥD the edge rDpD] and setΠ(rDpD])=S[rD,pD] ].

(T2) If Proposition 8.1 (a) applies, then the subgraph (D,D]+)G containsa pair of independent pD, qD-qD] , rD]-chains, both independent ofS[rD,pD] ]. We choose one such pair, say an s-t-chain X and an s′-t′-chain X ′, include the edges st and s′t′ in ΥD and set Π(st) = X andΠ(s′t′)=X ′.


(T3) Finally, if Proposition 8.1 (a) does not apply, then (D,D]+)G containsan x-y-chain for each x ∈ pD, qD and each y ∈ qD] , rD], all of themindependent of S[rD,pD] ]. We fix one such x-y-chain Px,y for each pairof x and y, include xy in ΥD, and set Π(xy)=Px,y. If A1≺D≺D]≺A2,there are four such edges xy, otherwise there are two.

Edges that fall under the item (T1) or (T2) will be called black while thoseunder the item (T3) will be called white. White edges within the same sub-graph ΥD will be said to form one class. Thus each class consists of eitherfour or – if D=A1 or D=A[2 – two white edges.

Observe that the graph Υ need not be uniquely determined because ofthe choice made in item (T2). Nevertheless, the choice has no effect on thepurpose for which Υ is constructed.

The following lemma explains why the edges of Υ have been classified asblack or white.

Lemma 8.2. Two chains Π(e) and Π(f) in M corresponding to distinctedges e and f of the transfer graph Υ are internally independent unless eand f are white edges from the same class.

Proof. Consider arbitrary two edges e=st and f=uv of Υ . We may clearlyassume that s ∈ ΘD, t ∈ ΘD] , u ∈ ΘE , and v ∈ ΘE] for some circuits Dand E with D E. By the definition of Υ , the chains Π(st) and Π(uv)are contained in (D,D]+)G and (E,E]+)G, respectively. If E D], then(D,D]+)G and (E,E]+)G have no edge in common, and so have Π(st) andΠ(uv). If E=D], the common edges of Π(st) and Π(uv) are contained inS∩(D,D]+)G∩(E,E]+)G=ΘD] =ΘE . In both cases, Π(st) and Π(uv) areinternally independent.

Let E =D and suppose that Π(st) and Π(uv) are not internally inde-pendent. Now both st and uv join ΘD to ΘD] , but none of them coincideswith rDpD] . Indeed, if we had, say, st=rDpD] , then Π(st)=S[rD,pD] ] whileΠ(uv) would be independent of S[rD,pD] ], as follows from (T2) and (T3);hence Π(st) and Π(uv) would be independent, a contradiction. Obviously,st and uv cannot fall under the item (T2) in the definition of Υ . The onlypossibility that remains is that they fall under the item (T3). In other words,they are white and belong to the same class.

Let ξ=v1v2 . . .vk be a path in Υ encoded by its vertex-sequence. We saythat ξ is increasing if for any two consecutive vertices vi∈ΘD and vi+1∈ΘD′of ξ one has D≺D′. Note that every increasing a1-a2-path in Υ traversesall the nests, so its length is the number of nests less one.

Consider an increasing path ξ=v1v2 . . .vk in Υ . Then for any two consec-utive edges vi−1vi and vivi+1 of ξ the corresponding chains Π(vi−1vi) and


Π(vivi+1) inM traverse their common edge vi in the same direction. Hencewe can form the walk Π(v1v2)tΠ(v2v3)t . . .tΠ(vk−1kt), which we denoteby Π(ξ). Clearly, Π(ξ) is alternating. As we show next, it is a chain.

Lemma 8.3. If ξ is an increasing path in Υ , then Π(ξ) is a chain in M.

Proof. Let ξ = v1v2 . . .vk be an increasing path in Υ . We already knowthat Π(ξ) is an alternating walk in M whose first and last edges are rays.To see that Π(ξ) is a chain, consider two constituting chains Π(vivi+1) andΠ(vjvj+1) corresponding to the edges vivi+1 and vjvj+1 of ξ with i<j. Let Dand E be circuits in Φ such that vi∈ΘD, vi+1∈ΘD] , vj∈ΘE , and vj+1∈ΘE] .Since ξ is increasing, we have D≺E. From the definition of Υ we further getthat Π(vivi+1) is contained in (D,D]+)G while Π(vjvj+1) is contained in(E,E]+)G. Hence Π(vivi+1) and Π(vjvj+1) are either disjoint or E=D] andtheir intersection is contained in S∩ (D,D]+)G∩ (E,E]+)G =ΘD] =ΘE . Inthe latter case, vi+1 is their only common edge inM, which in turn impliesthat Π(ξ) is a chain.

Our next result shows that in order to find two independent transversalchains in M it suffices to identify two suitable internally disjoint increasingtransversal paths in Υ .

Proposition 8.4. Let Υ be a transfer graph for a singular primitive 2-meshM with folded transversal wire. Let ξ and σ be two internally disjointincreasing a1-a2-paths in Υ which together use at most one white edge ofeach class. Then Π(ξ) and Π(σ) are independent transversal chains in M.

Proof. Let ξ= v1v2 . . .vt and σ=w1w2 . . .wt be two increasing a1-a2-pathsin Υ . Lemma 8.3 and the fact that ξ and σ connect a1 to a2 together implythat that Π(ξ) and Π(σ) are transversal chains. Suppose to the contrarythat Π(ξ) and Π(σ) share an F -edge e. Since ξ and σ are internally disjoint,e does not belong to any nest. Thus there exist indices i and j such thate is an inner edge of both Π(vivi+1) and Π(wjwj+1). By Lemma 8.2, thecorresponding edges vivi+1 and wjwj+1 of Υ are white and belong to thesame class. By our assumption, ξ and σ together use at most one whiteedge of each class, which is a contradiction. Thus Π(ξ) and Π(σ) are indeedindependent, as claimed.

It remains to show that a transfer graph Υ for M does contain twoincreasing transversal paths with the desired property. Before establishingthis fact we need one more lemma.

Lemma 8.5. Each component of the black subgraph of Υ −a1,a2 is apath traversing nests in a monotone order.


Proof. The reader may easily check that in Υ − a1,a2 each vertex isincident with at most one black edge leading forward and with at most oneleading backwards. This clearly establishes the lemma.

Now we are in position to prove a result that completes the argumentsfor Theorem 3.2.

Proposition 8.6. Let Υ be a transfer graph for a singular primitive 2-meshM with folded transversal wire. Then Υ has two increasing internallydisjoint a1-a2-paths which together use at most one edge of each class ofwhite edges.

Proof. Let Ψ = D ∈ Φ;D1 D D2 and ΥD contains white edges. Weregard Ψ as an ordered set with respect to the ordering inherited from M.

First suppose that Ψ = ∅. Thus Υ is all black. By the definition of Υ ,particularly the items (T1) and (T2), each ΥD consists of three black edgeswhich are independent whenever A1≺D≺D2. It follows that Υ consists ofthree increasing internally disjoint black a1-a2-paths, proving the statementwhen Ψ=∅.

Now let Ψ 6=∅. Take an arbitrary circuit D∈Φ, pick a vertex u∈ΘD, andlet E be the smallest element in Ψ such that ED. Lemma 8.5 implies thatin Υ there exists a unique increasing path from u to ΘE (possibly trivial)entirely consisting of black edges. Let us denote this path by β(u). Sincethere is no black edge between pE , qE and qE] , rE], the path β(u) doesnot extend into a longer increasing black path from u unless it terminatesat rE .

Let I1 and I2 be the smallest and the largest element of Ψ , respectively.By induction with respect to the ordering of Ψ we now establish the followingclaim.

Claim. For every circuit D ∈ Ψ with I1 D I2, the transfer graph Υcontains two disjoint increasing ΘD1-ΘD-paths which together use at mostone edge from each class of white edges. Furthermore, one of the paths beginsat pD1 and one of the paths (possibly the same) ends at rD.

To start the induction, consider the path β(pD1) whose terminal vertexis in ΘI1 . If the terminal vertex is rI1 , for the required pair of paths wechoose β(pD1) and, say, β(qD1). By the definition of β, these paths are blackand hence disjoint. By the choice of I1, they are ΘD1-ΘI1-paths, and byLemma 8.5 they are increasing. Thus they fulfil all the required properties,and the claim is verified in this case. If, on the other hand, the terminalvertex of β(pD1) is one of pI1 or qI1 , then, by Lemma 8.5, there exists aunique vertex d∈qD1 , rD1 such that β(d) ends at rI1 . In this case the tworequired paths are β(pI1) and β(d). The basis of induction is complete.


Now assume that the claim is true for some D ∈ Ψ with I1 D ≺ I2,and let γ and λ be increasing ΘD1-ΘD-paths such that one of them initiatesat pD1 and one of them, say λ, terminates at rD. Let d ∈ pD, qD be theterminal vertex of γ. Denote by E the smallest element of Ψ following D;clearly, D]E I2. We extend γ and λ to two ΘD1-ΘE-paths γ′ and λ′ asfollows. First of all, set λ′ = λβ(rD). If the terminal vertex of β(rD) is rE ,take γ′=γdrD]β(rD]). Otherwise, the terminal vertex of β(rD) is one of pEand qE . In this case let h ∈ qD] , rD] ⊆ΘD] be the vertex such that β(h)terminates at rE , and set γ′=γdhβ(h). Observe that in both cases we haveconstructed the paths γ′ and λ′ by extending γ and λ at their far ends insuch a way that one of the extended paths terminates at rE . Since one ofγ and λ starts at pD1 , so does one of γ′ and λ′. As above, it is easy to seethat the extended paths have all the required properties. This concludes theinduction step and establishes the claim.

To finish the proof, we take a pair γ and λ of ΘD1-ΘI2-paths guaranteedby the above claim and extend them to increasing a1-a2-paths as follows. Foreach vertex u∈ΘD1 there exists an edge a1u in ΥA1 . Thus we can add theblack edge a1pD1 to the path initiating at pD1 , and the edge a1v to the otherpath whose initial vertex we have denoted by v. At the other end we proceedas follows. Assume that λ is the ΘD1-ΘI2-path that terminates at rI2 , and letw be the terminal vertex of γ. Take the black edge rI2pI]2

and an arbitrary

edge from w to a vertex x 6=pI]2

in ΘI]2

; this edge will necessarily be white. By

Lemma 8.5, for each vertex y∈ΘI]2

there exists a unique maximal increasing

black path from y; it will cause no confusion if we denote it by β(y). SinceI2 is the largest element of Ψ , all these paths terminate at a2. Furthermore,a2 is their only common vertex. It follows that γwxβ(x) and λrI2pI]2

β(pI]2

)

are two internally disjoint increasing a1-a2-paths which together use at mostone white edge from each class. The proof is now complete.

Proof of Theorem 3.2. We want to see that every 2-mesh is good. Propo-sition 3.3 shows that this is indeed the case for every non-singular 2-mesh.For singular meshes we have proved in Proposition 4.1 that one can restrictto primitive 2-meshes. Proposition 6.2 shows that every primitive singu-lar 2-mesh with a smooth transversal wire is good. Finally, the fact that aprimitive singular 2-mesh with a folded transversal wire is good follows fromProposition 8.4 and Proposition 8.6.

Acknowledgements. Research reported in this paper was partially sup-ported by the project APVV-0223-10 and by the APVV project ESF-EC-0009-10 within the EUROCORES Programme EUROGIGA (project GRe-

94 E. MACAJOVA, M. SKOVIERA: SPARSELY INTERSECTING PERFECT . . .

GAS) of the European Science Foundation. The first author was also sup-ported by VEGA 1/1005/12.

Part of this work was done while its authors stayed at LaBRI, UniversiteBordeaux I. The authors wish to express their gratitude to Andre Raspaudfor hospitality and useful discussions.

The authors are indebted to a referee for his careful reading and usefulsuggestions which improved the presentation of this paper.

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Edita Macajova, Martin Skoviera

Department of Computer Science

Faculty of Mathematics, Physics and Informatics

Comenius University

842 48 Bratislava, Slovakia

macajova,[email protected]

mailto:[email protected]

mailto:[email protected]

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Sparsely intersecting perfect matchings in cubic graphs