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mJournal of Combinatorial Theory, Series B � 1652

journal of combinatorial theory, Series B 66, 1�10 (1996)

A Class of Matchings and a Related Lattice

K. Kilakos

Laboratoire de Combinatoire et d'Informatique Mathe� matique,De� partement de Mathe� matiques et d'Informatique,

Universite� du Que� bec a� Montre� al, Montre� al, Canada

Let '$(G) denote the fractional chromatic index of a graph G. Let M(G) be afamily of matchings of G such that M # M(G) if and only if there exists a fractional'$(G)-edge colouring of G in which M is a colour class. We use Lova� sz's matchinglattice theorem to obtain a description of the lattice generated by M(G). We shallshow that, in a sense, the only complication arises when G has the Petersen graphminus a vertex as a minor. � 1996 Academic Press, Inc.

1. Introduction

The lattice of perfect matchings of a graph G=(V, E) has been charac-terized by Lova� sz [6]. He has shown that when every edge of G lies on aperfect matching, then a set of obvious necessary conditions for a vectorw # ZE(G) to belong to this lattice is also sufficient provided that after aseries of contractions we do not obtain the Petersen graph. Thus, as notedby Lova� sz, ``the Petersen graph is, in a sense, the only difficult example.''Here we use Lova� sz's result to characterize a related lattice. To make itsnotion precise, we need the following definitions.

Let G be a graph and J (G) denote the family of matchings of G. Anedge colouring of G is a partition of E(G) into matchings or, in terms ofvectors, any point in

P$I (G) :={ y # [0, 1]J(G): :e # M # J(G)

yM=1, e # E(G)= . (1)

Let 1� be the row vector of all 1's. The chromatic index of G, denoted by/$(G), is given by

min[1� } y: y # P$I (G)]. (2)

A fractional edge colouring is any point in

P$(G) :={ y # (Q & [0, 1])J(G): :e # M # J(G)

yM=1, e # E(G)= , (3)

article no. 0001

10095-8956�96 �12.00

Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

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and the fractional chromatic index '$(G) of G is given by

min[1� } y: y # P$(G)]. (4)

G is k-edge colourable (resp. fractionally k-edge colourable) if the minimumin (2) (resp. (4)) is at most k.

The family M(G) is defined to contain precisely those matchings ofJ (G) which in some fractional '$(G)-edge colouring y index a strictlypositive component. Our goal is to characterize the lattice generated byM(G), denoted by lat(M(G)). We shall show that, in a sense, the onlydifficult case arises when after a series of contractions, we obtain a graphon nine or ten vertices which has the Petersen graph minus a vertex as asubgraph.

The study of various objects associated with matchings of graphs hasbeen motivated to a large extent by the theory of edge colourings. Anr-graph is an r-regular graph in which for every subset of vertices of oddcardinality there are at least r edges with exactly one end in this subset.One of the earliest problems in edge colourings is Tutte's conjecture [8]:

Conjecture 1.1. Every 3-graph with no Petersen graph as a minor is3-edge colourable.

This conjecture is still unresolved. However, Seymour [7] has shownthat if we drop the non-negativity constraint in (1) (that is, let y # ZJ(G)),then Tutte's conjecture holds. Lova� sz has since generalized this result. Bycharacterizing the lattice of perfect matchings of any graph, he was able toshow that such a relaxation could also be proved for the followinggeneralization of Tutte's conjecture, due to Seymour [7]:

Conjecture 1.2. Every r-graph that does not have the Petersen graph asa minor is r-edge colourable.

Our characterization of the lattice generated by M(G) also proves theabove relaxations. In addition, it gives us analogous relaxations of thefollowing two conjectures, the first due to Goldberg [2] and Seymour [7],and the second to Kilakos and Shepherd [5].

Conjecture 1.3. For every graph G, /$(G)�max[2+1, W'$(G)X],where 2 denotes the maximum degree of the graph.

Conjecture 1.4. Every graph that does not have the Petersen graphminus a vertex as a minor is W'$(G)X-edge colourable.

Ideally, one would like to have a description of the cone generated byM(G). Such a result would, of course, lead to the resolution of the aboveconjectures. However, this might be impossible to obtain. Holyer [3] has

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shown that deciding 3-edge colourability of cubic graphs is an NP-com-plete problem.

We conclude this section with a few notational conventions. Let S be asubset of vertices of a graph G. We denote by $G(S), or simply $(S), theset of edges of G with exactly one end in S and by E(S) the set of edgesof G with both ends in S. For T�V(G), we denote by S�T the set(S"T ) _ [v] where v is a new vertex. The graph induced by S�T, G[S�T],is the graph obtained from G[S _ T] by contracting G[T] to a singlevertex. Let F�E(G) and F $�F. Let w be a vector indexed by F. Therestriction of w in F $ is the |F $|-dimensional vector whose componentscorrespond to the components of w indexed by F $. We denote by w(F $) thenumber �e # F $ we .

2. The Lattice Generated by Mr(G)

We shall find it convenient to present our theorem in terms of Mr(G), thefamily of matchings of G whose elements are precisely those M # J (G) forwhich there exists a fractional r-edge colouring y, such that yM>0. Webegin by presenting some of the preliminary material needed in the descrip-tion of lat(Mr(G)).

A connected graph G in which every edge is contained in a perfectmatching is called a matching covered graph. An edge cut $(S) of G,S/V(G), is tight if every perfect matching of G contains exactly one edgefrom it. Note that if $(S) is tight, then S is of odd cardinality. Trivial edgecuts are of the form $(v), for some v # V(G). The proof of our theorem isbased on the following results of Lova� sz.

Theorem 2.1. Let G be a matching covered graph that has no non-trivialtight cuts and is different from the Petersen graph. Let w # ZE(G). Then w isin the lattice generated by the perfect matchings of G if and only if w($(u))is the same value for each vertex u # V(G).

The Petersen graph adds an extra complication.

Theorem 2.2. Let P be the Petersen graph and w # ZE(P). Then wbelongs to the lattice of perfect matchings of P if and only if w($(u)) has thesame value for every u # V(P) and for every 5-circuit C of P, w(E(C)) is aneven number.

We should note that the complete version of Lova� sz's result deals withall matching covered graphs, by combining the above theorems through aseries of contractions along tight cuts.

3a class of matchings

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The fractional edge colouring problem is a well-solved problem. Edmonds'matching polyhedron theorem [1] implies that for every graph G,

'$(G)=max {[2] _ { 2 |E(U)|( |U|&1)

: U�V(G), |U|>1 and odd== . (5)

This equality motivates the following definition. A subset S of vertices ofG is called r-full, or simply full when there is no ambiguity, if S is of oddcardinality and |E(S)|=r�(( |S|&1)�2). Note that a single vertex is an r-fullset, a trivial one.

We make an observation regarding (5).

Lemma 2.3. M # Mr(G) if and only if for every r-full set S of G,|M & E(S)|=( |S|&1)�2) and for every vertex u of degree r, |$(u) & M|=1.

As in Lova� sz [6], the proof of the main theorem is based on a decom-position procedure. Let G be a graph and fix r as above. A full set of G isreducible if 3�|S|�|V(G)|&2. The decomposition procedure proceeds byfirst splitting the graph, in the obvious manner, along 1-cuts. Subsequently,we choose one of the resulting graphs which has a reducible full set S andobtain two graphs G1 and G2 by contracting G[S] and G&S, respectively,to a single vertex. We repeat until none of the resulting graphs has areducible full set.

An extended Petersen graph is an r-regular graph whose underlined simplegraph is the Petersen graph and |$(S)|>r for every non-trivial odd cardinalitysubset of vertices S. For any given r, the family of graphs Pr consists of thegraphs which can be obtained from an extended Petersen graph by deletingeither a vertex or zero or more edges incident to the same vertex.

Theorem 2.4. Let G be a connected graph and w # Z(E(G). Let r�'$(G),and fix a decomposition of G. Then w # lat(Mr(G)) if and only if thefollowing conditions hold.

(a) For every r-full set S used in the decomposition or appears in anon-further decomposable graph w(E(S))=p(( |S|&1)�2) and for everyvertex u of degree r of G, w($(u))=p, for some p # Z.

(b) For every H # Pr which is the result of the decomposition and anygiven full set S of V(H), the restriction of w on any 5-circuit of H[S] sumsup to an even number.

Proof. We may assume r # Z, since there exists x # Z so that x'$(G) # Z,and we can add to each edge x&1 parallel edges, each with w value equalto zero. Suppose that w # lat(Mr(G)). Then it is immediate from Lemma 2.3that (a) holds. Also, (b) holds as well because any matching M # Mr(G) issuch that |M & E(C)|=0 or 2 for every 5-circuit C of H[S].

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Conversely, if G is not 2-connected, then by splitting G along a vertexcut into two graphs G1 and G2 , one can easily deduce that w # lat(Mr(G))if and only if its restriction in G1 belongs to lat(Mr(G1)) and its restrictionin G2 belongs to lat(Mr(G2)). Thus, we may assume that G is 2-connected.Our analysis depends on whether or not G has reducible full sets.

Case 1. G has no reducible full sets. We distinguish two subcases.Assume first that the only full sets of G are the trivial ones. Let G$ and

w$ # ZE(G$) be a duplicate copy of G and w respectively. We constructa graph G� and a vector w~ # ZE(G� ) as follows. V(G� )=V(G) _ V(G$) andE(G� )=E(G) _ E(G$) _ X, where X is the set of edges obtained from joiningeach vertex v # V(G) to its image v$ # V(G$) with r&|$G(v)| edges. Now w~is such that its restriction in E(G) (resp. E(G$)) is w (resp. w$) and for everyvertex u # V(G� ), w~ ($G� (u))=p, where p could be any integer if G has novertices of degree r.

The following is immediate from Lemma 2.3.

2.5. M # Mr(G) if and only if it is the restriction in G of a perfectmatching of G� .

Thus, if w~ belongs to the lattice of perfect matchings of G� , then clearlyw belongs to lat(Mr(G)). Observe also that the underlying simple graph ofG� is not the Petersen graph. Hence, in view of Theorem 2.1, it would sufficeto show that the only tight cuts of G� are the trivial ones.

Let S� be an odd cardinality subset of V(G� ), 3�|S� |�|V(G� )|&3. LetT=S� & V(G) and R$=S� & V(G$). Let T $ be the image of T in G$ and R theimage of R$ in G. Without loss of generality, |T"R| is odd. Denote by t thenumber of edges of G� with ends in both V(G� )"(T _ R) and T & R. Observethat

|$G� (S� )|�|$G� (T"R)|+|$G� (R"T )|+t.

Since the only full sets of G are the trivial ones, |$G� (T"R)|>r when|T"R|>1. When |T"R|=1, since G is 2-connected, one of the other twoquantities in the above inequality is different from zero. Thus, |$G� (S� )|>r.Also, since G� is r-regular and, by (2.5), fractionally r-edge colourable, eachmember of Mr(G� ) is a perfect matching. Thus, there must be a perfectmatching M� of G� such that |M� & $G� (S� )|>1, as required.

Next assume that G has a non-reducible full set S (i.e., a full set such that|V(G)|&|S|�1). Let now G� and w~ # ZE(G� ) be obtained, respectively, fromG and w as follows. If G$G[S] then V(G� )=V(G) _ [v], otherwiseV(G� )=V(G). Let v be the vertex of V(G� )"S and join v to every vertex u # Swith r&|$G(u)| edges. By using the fact that for each vertex u of degree rof G, w($G(u))=p and w(E(S))=p(( |S|&1)�2), one can easily deduce that

5a class of matchings

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w~ can be defined so that w~ ($G� (u))=p, for every u # V(G� ), and its restrictionin G equals w.

Note that by its definition and by the assumption that S is a full set,G� is r-regular. Note also that for every odd cardinality S� �V(G� ),3�|S� |�|V(G� )|&3, |$(S� )|>r since G has no reducible full set and eitherS� or V(G� "S� ) is a subset of V(G). Thus, G� is fractionally r-edge colourablewith no non-trivial tight cuts. Moreover, (2.5) also holds here. Thus, wecan again conclude, by Theorem 2.1, that w # lat(Mr(G)) when G � Pr .When G # Pr , the result follows from Theorem 2.2 and the fact that everyM # Mr(G[S]) (resp. every perfect matching M� of G� ) intersects every5-circuit C of G[S] (resp. C of G� ) in either zero or two edges.

Case 2. G has a reducible full set S. Let G1 and G2 be respectively thegraphs resulting from contracting G[S] and G&S to a single vertex. Letw1 and w2 be the restriction of w in G1 and G2 respectively. It suffices toshow that G1 and G2 are fractionally r-edge colourable and w # lat(Mr(G))if and only if w1 # lat(Mr(G1)) and w2 # lat(Mr(G2)).

Since S is full, by Lemma 2.3, for every M # Mr(G), |$(S) & M|�1. Thisimplies that the restriction of M in G1 and G2 , respectively, belongs to Mr(G1)and Mr(G2). It follows that G1 and G2 are fractionally r-edge colourableand if w # lat(Mr(G)), then w1 # lat(Mr(G1)) and w2 # lat(Mr(G2)).

Suppose now that w1 # lat(Mr(G1)) and w2 # lat(Mr(G2). Then, there existweight vectors yi # ZE(Gi), i=1, 2, such that �M # Mr(Gi)

yiM/M=wi, where

/M is the incidence vector of M. Using y1 and y2 we shall derive a familyF of matchings of G and a vector z # ZF as follows.

First, for each edge e # $(S) repeat the following procedure until theweights of the matchings in Mr(G1) and Mr(G2) that contain e are all zero.Let M1 # Mr(G1), M2 # Mr(G2) be such that e # M1 , M2 . If M1 and M2 canbe chosen so that y1

M1, y2

M2{0, then let M1 _ M2 # F, zM1 _ M2

= y1M1

, sety1

M1=0 and reduce y1

M2by y1

M1. Otherwise, if only, say, y1

M1{0, then select

a second matching N1 # Mr(G1) such that e # N1 and y1N1

{0, let M1 _ M2 ,N1 _ M2 # F, zM1 _ M2

= y1M1

, zN1 _ M2= y1

N1, set y1

M1=0, y1

N1=0 and reduce

y2M2

by y1M1

+ y1N1

. (Note that since �e # M # Mr(G1) y1M=�e # M # Mr(G2) y2

M , N1

can be selected as specified). Clearly, this procedure terminates.Next, we deal with the matchings of Mr(G1), Mr(G2) that do not contain

an edge e # $(S). If both G1 and G2 contain non-trivial full sets or verticesof degree r, then, as above, pair off matchings of Mr(G1) with matchingsof Mr(G2) that do not contain an e # $(S), until all the weights of thematchings in Mr(G1) _ Mr(G2) have been reduced to zero. This is alwayspossible since for every full set R # V(Gi), i # [1, 2], wi(E(R))=p(( |R|&1)�2) and for every vertex u # V(Gi) of degree r, w($Gi (u))=p. If,say, G2 has only trivial full sets and no vertex of degree r, then pair off suchmatchings until the weight of any matching in Mr(G1) is reduced to zero.

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(This is always possible by choosing at each step an M1 # Mr(G1) such thatyM1

{0). For any remaining matching M # Mr(G1), with yM{0, let M # Fand zM= y1

M .Observe that for each e # E(G), �e # M # F zM=w(e). Let yM=zM for

each M # Mr(G) & F and let yM=0 for any M # Mr(G)"F. In order toconclude the proof of the theorem, it suffices to show that F�Mr(G). Weshall need the following ``uncrossing'' lemma which can be understoodindependently of the theorem.

Lemma 2.6. Let G be a graph and r # Z, r�W'$(G)X. Let S, T be fullsets of G.

(a) If T�S, then S�T is full.(b) If |S & T | is odd, then S & T and S _ T are full.(c) If |S & T | is even, then S"T and T"S are full.

Proof. (a) It follows from a straightforward counting of the edges ofG[S�T].

(b) |E(S _ T)|�|E(S)|+|E(T )|&|E(S & T )|. Also,

|E(S _ T )|+|E(S & T )|�r|S _ T |&1

2+r

|S & T |&12

.

Thus, S _ T and S & T are full.

(c) This is an adaptation of a result in [4].

|E(S"T )|+|E(T"S)|=|E(S)|+|E(T )|&2 |E(S & T )|&|$(S & T )|

=r|S|&1

2+r

|T |&12

&2 |E(S & T)|& |$(S & T)|

=r|S"T |&1

2+r

|T"S|&12

+2r|S & T |

2

&2 |E(S & T )|&|$(S & T )|

�r|S"T |&1

2+r

|T"S|&12

.

Also,

|E(S"T )|+|E(T"S)|�r|S"T |&1

2+r

|T"S|&12

.

Thus, S"T and T"S are full. K

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Let M # F. It is clear that for every vertex u of degree r of G,|$(u) & M|=1. Let T be a full set of G. If T�S or T & S=<, then|M & E(T )|=(( |T |&1)�2). So assume that T�3 S and T & S{<. FromLemma 2.6 we have the following.

If |T & S| is odd, then the fact that (S _ T)�S and T�(S & T ) are fullimplies that there is no edge with ends in both S"T and T"S. Moreover,because S & T and (S _ T )"S are full, |M & E((S _ T )�S)|=(( |(S _ T )�S|&1)�2), and |M & E(S & T )|=(( |S & T |&1)�2). It follows that|M & E(T )|=(( |T |&1)�2).

If |T & S| is even, then T�(T"S) is full and because G1 is fractionallyr-edge colourable, (V(G)"(S"T ))�(V(G)"S) is also full and thus, thereis no edge with ends in both S & T and V(G)"(S _ T). In addition,|M & E(T"S)|=((|T"S|&1)�2) and |M & E(T�(T"S))|=((|T�(T"S)|&1)�2)because T"S and T�(T"S) are full. Again, it follows that |M & E(T)|=(( |T |&1)�2). Thus, in either case, M # Mr(G), as required. This completesthe proof of the theorem. K

We conclude this paper with a theorem regarding the decompositionprocedure. The proof technique we employ is borrowed from Lova� sz whoproved an analogous result for tight cut decompositions of matchingcovered graphs.

From Lemma 2.6 one can easily deduce that for any given decompo-sition of a graph G, there is a one to one correspondence between thereducible full sets chosen and a maximal family F of such sets of G suchthat for any S, T # F, S"T=< and T"S=<. Thus, alternatively, one cana priori select such a family in G and proceed with the prescribed contrac-tions. The following theorem shows that the end result is independent ofthe specific decomposition used.

For our purposes, two graphs H and H$ are the same if they areisomorphic (including edge multiplicity) except that possibly, the edgesincident to a certain vertex v # V(H) and the edges incident to its imagev$ # V(H$) have different multiplicities when |V(H)| (=|V(H$)| ) is even.

Theorem 2.7. The result of any two decomposition procedures is thesame list of graphs.

Proof. We proceed by induction on |V(G)|. Let F and F$ be any twofamilies of reducible full sets corresponding to two decomposition proce-dures. Without loss of generality, we shall assume that for any S # F andS$ # F$, S${V(G)"S (for we can replace S$ by S).

Case 1. F and F$ have a member S in common. Then we can startboth lists by reducing along $(S). The result now follows from theinduction hypothesis applied to the resulting two graphs.

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Case 2. There is a reducible full set T such that either T"S=< orS"T=< and either T"S$=< or S$"T=<, for some S # F and S$ # F$.(T need not necessarily be distinct from S or S$). Consider any maximalfamilies H and H$ of reducible full sets of G that contain S, T and S$, Trespectively. By case 1, the decompositions corresponding to F, H andF$, H$ respectively, yield the same list of graphs. This is also true for Hand H$. It follows that F and F$ yield the same list of graphs.

Case 3. There exists S # F and S$ # F$ such that |S"S$|=|S$"S|=1.Let v and v$ be respectively the elements of S"S$ and S$"S. Observe thatbecause S and S$ are full, |[u: vu # E(G)] & (S & S$)|=|[u: v$u # E(G)] &

(S & S$)|. This implies that the two graphs obtained from reducing along$(S) are isomorphic to the graphs obtained from reducing along $(S$).Again the result follows by induction.

Case 4. There exists S # F and S # F$ such that |S & S$|=1 and S, S$are respectively the only members of F, F$. By Lemma 2.6, S _ S$ is fulland since F and F$ are maximal, we must have that |S _ S$|=|V(G)| or|V(G)|&1. Moreover, by Lemma 2.6, no edge of G has one end in S"S$and the other S$"S. It is now obvious that by contracting along $(S) and$(S$) we obtain the same graphs.

To conclude, we note that it is easy to check, with the help of Lemma2.6, that one of above cases must have occurred. K

Acknowledgments

Very special thanks to U.S.R. Murty for introducing me to the subject of matchingsand lattices. Many thanks also to the anonymous referee for many useful comments andsuggestions.

References

1. J. Edmonds, Maximum matching and a polyhedron with 0, 1 vertices, J. Res. Nat. Bur.Stand. B 69 (1965), 125�130.

2. M. K. Goldberg, On multigraphs with almost maximal chromatic class, Discret Analiz 23(1973), 3�7. [in Russian]

3. I. Holyer, The (NP)-completeness of edge-colouring, SIAM J. Comput. 10 (1981),718�720.

4. K. Kilakos and B. Reed, Fractionally colouring total graphs, Combinatorica 13 (1993),435�440.

5. K. Kilakos and B. Sphepherd, ``Minors and the Chromatic Index of r-graphs,'' TechnicalReport 93-62, DIMACS, 1993.

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10 k. kilakos