Minors and the Chromatic Index of r-graphs K. Kilakos B. Shepherd Dept de Math?ematique et d'Informatique , Mathematics and Operational Research , Universit?e du Queb?ec ? a Montr?eal , London School of Economics , Canada . U.K. Abstract . Let T2 be the graph obtained from the Petersen graph by first deleting a vertex and then contracting an edge incident to a vertex of degree two . We give a simple characterization of the graphs that contain no minor isomorphic to T2 . This characterization is used to show that if every planar r-graph is r-edge colourable , then every r-graph with no minor isomorphic to T2 is r-edge colourable . DIMACS Report No. 93-62 1 Introduction We consider lopless graphs G = -LRB- V ; E -RRB- with vertex set V and edge set E . A minor -LRB- respectively subgraph ; respectively induced subgraph -RRB- of G is any graph obtained from G by deleting vertices and contracting or deleting edges -LRB- respectively deleting vertices and edges ; respectively deleting vertices -RRB- . For a graph H , G is said to be H-free if no minor of G is isomorphic to H. For S ae V -LRB- G -RRB- , we denote with ffiG -LRB- S -RRB- , or simply ffi -LRB- S -RRB- , the set of edges of G with exactly one end in S . An r-graph is an r-regular graph in which jffi -LRB- S -RRB- j > = r for every subset of vertices S with odd cardinality . Figure 1 : The Petersen graph The subject of this paper is the following 2 conjectures . Conjecture 1.1 Every planar r-graph is r-edge colourable . Conjecture 1.2 Every r-graph with no Petersen graph -LRB- see Figure 1 -RRB- as a minor is r-edge colourable . Historically , Conjectures 1.1 and 1.2 date back to the famous four colour problem . Consider the special case when r = 3 . Then Conjecture 1.1 is Tait 's -LSB- 12 -RSB- equivalent formulation of the four colour problem and Conjecture 1.2 is Tutte 's -LSB- 14 -RSB- generalization of the same problem . Now to obtain Conjectures 1.1 and 1.2 in their general form we need only introduce the following conjecture of Seymour -LSB- 9 -RSB- : when r > = 4 , every r-graph G has a matching M such that G ? M is an -LRB- r ? 1 -RRB- - graph . This together with Conjecture 1.2 would imply the following conjecture attributed by Goddyn to Lov?asz . Conjecture 1.3 The perfect matchings of a P10-free graph form a Hilbert basis . The set M of perfect matchings of a graph G forms a Hilbert basis if each integer vector in the cone -LRB- fOM : M 2 Mg -RRB- is expressable as a nonnegative integer combination of incidence vectors of perfect matchings . There is a more general -LRB- than 1.1 -RRB- conjecture of Seymour -LSB- 10 -RSB- that also contains the four colour problem as a special case . Denote by ? the maximum degree of a graph G and let ? 0 -LRB- G -RRB- = maxff ? ; f2jE -LRB- S -RRB- j = -LRB- jSj ? 1 -RRB- : S ae V -LRB- G -RRB- ; jSj = 2k + 1 ; k > = 1gg , where E -LRB- S -RRB- = fvw 2 E -LRB- G -RRB- : v ; w 2 Sg . Sometimes ? 0 -LRB- G -RRB- is called the fractional chromatic index of G . It follows from the work of Edmonds -LSB- 2 -RSB- that ? 0 -LRB- G -RRB- = r when G is an r-graph and so Theorem 1.1 would imply that for r-regular graphs , fractional r-edge colourability implies that G is r-edge colourable . In the case that G is not r-regular things become more complex although there is a simple description of the fractionally 3-edge colourable graphs : a 2-connected graph G satisfies ? 0 -LRB- G -RRB- = 3 if and only if it arises by making k subdivisions of edges in a 3-graph , where k is any integer other than one . Gr?otszch has conjectured the following : Conjecture 1.4 -LRB- Gr?otzsch -RRB- For a planar graph G , ? 0 -LRB- G -RRB- = 3 implies G is 3-edge colourable . This conjecture would imply that the 3-edge colourability of a planar graph could be determined by finding its fractional chromatic index -LRB- which amounts to solving a linear program -RRB- . Seymour has conjectured that such a \ fractional determines integral '' property holds for all planar graphs : Conjecture 1.5 -LRB- Seymour -RRB- Every planar graph G is d ? 0 -LRB- G -RRB- e-edge colourable . This conjecture appears to require substantially more than the veracity of the four colour conjecture . It was only recently shown , for instance , that the result holds for series-parallel graphs -LRB- see -LSB- 11 -RSB- -RRB- . The smallest example of a graph not being d ? 0 -LRB- G -RRB- eedge colourable is the Petersen graph minus a vertex , denoted by T3 . This can also be described as K3 ; 3 with the edges of a perfect matching subdivided once . It was shown in -LSB- 4 -RSB- , that every T3-free graph G with ? 0 -LRB- G -RRB- = 3 is 3-edge colourable if Conjecture 1.4 holds -LRB- and that any T3 3-graph is 3-edge colourable -RRB- . This motivates us to propose the following strengthening of Conjecture 1.5 which would characterize the largest possible minor-closed class of graphs with the fractional-determines-integral property . Conjecture 1.6 Every T3-free graph is d ? 0 -LRB- G -RRB- e-edge colourable . We mention that Conjectures 1.5 and 1.6 also deal with the structure of the matching polyhedron of the corresponding graphs . The results of Edmonds -LSB- 2 -RSB- and Baum and Trotter Jr -LSB- 1 -RSB- imply that every replication -LRB- i.e. , adding multiple copies of edges -RRB- H of a subgraph of G is d ? 0 -LRB- H -RRB- e-edge colourable if and only if the matching polyhedron of G has the integer decomposition property -LRB- a polyhedron P has the integer decomposition property if , for each integer k and each integral vector z 2 kP -LRB- : = fkx : x 2 Pg -RRB- , z can be expressed as the sum of k integral vectors in P ; in particular P has integral extreme points -RRB- . Thus 1.6 could be stated as : the matching polyhedron of every T3-free graph has the integer decomposition property . The present paper attempts inroads to Conjectures 1.1 and 1.6 by considering the class of graphs with no minor isomorphic to the graph obtained from T3 by suppressing a degree two vertex . This graph is denoted by T2 -LRB- see Figure 3 -RRB- and note that it is obtained from K3 ; 3 by subdividing two independent edges . We give a decomposition theorem for T2-free graphs based on vertex 1 - and 2 - cuts and replicating degree three vertices -LRB- see Theorem 3.4 -RRB- . The decomposition is subsequently used to show that if Conjecture 1.1 is true , then every T2-free graph is r-edge colourable -LRB- see Theorem 4.1 -RRB- . 2 A Decomposition of T2-Free Graphs In this section we state the main decomposition result concerning the structure of T2-free graphs . Clearly , a graph is T2-free if and only if its underlying simple graph is T2-free . Thus we restrict ourselves to simple graphs . We note next that if G is a T2-free graph and v is a degree three vertex of G , then replacing v by a stable set of vertices with the same neighbourhood as v is again a T2-free graph . Moreover , if G is a planar graph -LRB- and therefore abviously T2-free -RRB- , we claim that by adding a single new vertex v0 adjacent to N -LRB- v -RRB- -LSB- fvg , also results in a T2-free graph . To see this , observe that if G has a T2 minor H , then ss -LRB- H -RRB- has to contain v and v0 . Moreover , in ss -LRB- H -RRB- , one of v , v0 has degree two and they are either adjacent or they are both adjacent to a vertex of degree three . But then , it is easy to see that we can construct a minor H ? = K3 ; 3 whose expansion does not contain one of v , v0 . This clearly contradicts the fact that G is planar . The above discussion essentially describes the decomposition of T2-free graphs . Since composing along 1 and 2-cuts preserves T2-freeness , this yields a decomposition procedure for the class of T2-free graphs . We now introduce two classes of graphs in order to give the precise statement of the theorem . Let F be a graph . Two vertices v , v0 of F are mates if in G ? vv0 they are both of degree three and N -LRB- v -RRB- = N -LRB- v0 -RRB- . Let I be a stable set of degree three vertices of G. Let G be a graph obtained from F by adding for each v 2 I either a mate v0 of v and the edge vv0 or by adding a non-empty set of mutually non-adjacent mates of v. Let M be the set of mate vertices so added . We say that G is a mate graph of F and write G = -LSB- F ; I ; M -RSB- . Note that every graph can be construed as a mate graph of some graph since G = -LSB- G ; ; ; ;] . We shall denote by M the class of mate graphs of planar graphs . A graph G 2 K if either -LRB- a -RRB- has a stable set I such that G ? I is a subgraph of K4 , or -LRB- b -RRB- has a stable set I consisting of pairwise non-adjacent replications of at most two vertices -LRB- respectively of a single vertex -RRB- such that , G ? I is a subgraph of K5 -LRB- respectively of K6 -RRB- -LRB- see figure 2 -RRB- . K K K 4 5 6 Figure 2 : Maximal members of K Theorem 3.4 Let G be a 3-connected graph , jV -LRB- G -RRB- j > = 9 . G is T2-free if and only if G 2 M -LSB- K. Robertson -LSB- 8 -RSB- and Kelmans -LSB- 3 -RSB- have independently obtained decomposition results for the related class of triangle-free graphs with no M -LRB- 4 -RRB- minor . -LRB- A M?obius 4-ladder , M -LRB- 4 -RRB- , is a circuit -LRB- v0 ; ? ? ? ; v7 -RRB- with the edges vivi +4 , i = ; ? ? ? ; 3 . -RRB- Kelmans has shown that for 3-connected graphs G with no triangles such that , for every 3-cut S , G ? S has exactly two components one of which is a single vertex , then G does not have a M?obious 4-ladder as a minor if and only if it is planar or a bipartite graph whose one side of the bipartition has at most four vertices . This result has also been obtained by Robertson who in addition showed how G can be decomposed along 3-cuts so that G is M -LRB- 4 -RRB- - free if and only if -LRB- both -RRB- the resulting graphs are M -LRB- 4 -RRB- - free . The statement of Theorem 3.4 is distinct from these results in two ways . The first is that it considers T2 which is a proper subgraph of the M?obius 4-ladder and hence a more restrictive excluded minor but on the other hand it does not disallow triangles . The other distinction is that it considers only a special type of 3-cut reduction -LRB- those arising from degree three replications -RRB- . This is because for our purposes we ideally Figure 3 : The graph T2 seek operations which can be used to reduce associated edge-colouring problems . It turns out that degree three replications can be used in this way for the edge-colouring problems on r-graphs -LRB- see Theorem 4.1 -RRB- . Such a decomposition for T3-free 3-graphs is simple and is given in -LSB- 4 -RSB- . For general graphs , however , finding such colouringpreserving operations seems to be a difficult problem ; it was only recently shown , for example , how to reduce edge-colouring problems -LRB- i.e. , with a view to resolving Conjecture 1.5 -RRB- along vertex 2-cuts -LSB- 6 -RSB- . We close with few remarks about minors and notation . If H is a maximum degree three minor of a graph G , then G contains a subdivision of H as a subgraph . Thus , there exists a one-to-one mapping f which makes correspond with each vertex v of H a vertex f -LRB- v -RRB- of G , and a one-to-one mapping ss which makes correspond with each edge e = uv of H a path ss -LRB- e -RRB- of G connecting f -LRB- u -RRB- and f -LRB- v -RRB- . Moreover , for any two edges y1y2 , x1x2 , V -LRB- ss -LRB- y1y2 -RRB- -RRB- \ V -LRB- ss -LRB- x1x2 -RRB- -RRB- = ff -LRB- y1 -RRB- ; f -LRB- y2 -RRB- g \ ff -LRB- x1 -RRB- ; f -LRB- x2 -RRB- g . It is convenient to assume that the vertices of a minor H have the same labels as the corresponding vertices of G . In other words , for v 2 V -LRB- H -RRB- , f -LRB- v -RRB- = v. Thus , if U is a subset of V -LRB- G -RRB- , and H is a cubic graph with V -LRB- H -RRB- = U , then H is a minor of G if for each edge uv of H there corresponds a -LRB- u ; v -RRB- - path ss -LRB- uv -RRB- of G , called the expansion of uv , such that , for any two edges x1x2 , y1y2 of H , V -LRB- ss -LRB- x1x2 -RRB- \ ss -LRB- y1y2 -RRB- -RRB- = fx1 ; x2g \ fy1 ; y2g . Let H be a minor of a graph G . An edge e of H is said to be short if ss -LRB- e -RRB- has exactly one edge ; otherwise e is long . For a subgraph F of H we denote by ss -LRB- F -RRB- the graph -LSB- e2F ss -LRB- e -RRB- . A subgraph L of G is H-external , if its edge set is contained in E ? ss -LRB- H -RRB- . When L is a path with ends v and w and V -LRB- L -RRB- -LSB- V -LRB- ss -LRB- H -RRB- = fv ; wg -RRB- , then L is also called an H-external path of G . The attachment vertices of L are the elements of V -LRB- L -RRB- \ V -LRB- ss -LRB- H -RRB- -RRB- . The attachment edges of L are the edges of H whose expansions contain an attachment vertex of L. 3 Proof of the Decomposition Theorem Recall that M denotes the class of mate graphs of planar graphs . We commence with a characterization of the class M , but first we verify the following simple property of these graphs . 3.0.1 M is closed under taking subgraphs . Proof . Let G = -LSB- F ; I ; M -RSB- , where F is a planar graph , and let e 2 E -LRB- G -RRB- . Note that if e is not incident to any vertex of I or M , then G ? e = -LSB- F ? e ; I ; M -RSB- . So suppose that e is incident to a vertex v0 which , without loss of generality , is not in I but is a mate of a vertex v in I. Firstly , if e = vv0 , then , in G ? e , v and v0 are non-adjacent mates and so G ? e = -LSB- F ; I ; M -RSB- . So we assume that e < > vv0 . Let x1 , x2 , x3 be the neighbours of V in G ? vv0 . We may assume that e = v0x1 . Let F be the subgraph of G with V -LRB- F -RRB- = V -LRB- F -RRB- -LSB- fv0g and E -LRB- F -RRB- = E -LRB- F -RRB- -LSB- -LRB- ffi -LRB- v0 -RRB- nfv0x1g -RRB- . Clearly , G ? e = -LSB- F ; I ; M -RSB- , where I = I or Infvg and M = Mnfv0g . But since v has degree three in F , x2 , x3 lie on the boundary of a face of F , and so F is also planar . It follows that G ? e 2 M. 2 Lemma 3.1 Let G be a 3-connected graph . G 2 M if and only if every K3 ; 3 minor H of G has two mate vertices of G. Proof . Suppose that G 2 M and let H be a K3 ; 3 minor of G. Clearly , ss -LRB- H -RRB- 2 M , and therefore is obtained from a planar graph by replicating a stable set I . If G is obtained from a planar graph by replicating some stable set I , then the proof of -LRB- 3.0.1 -RRB- indicates that I can be chosen to be a subset of I . Since I can only contain degree three vertices , we conclude that H has two mate vertices of G. Conversely , write G = -LSB- F ; I ; M -RSB- , where F has as few vertices as possible . We shall show that F is a planar graph . Suppose this is not true . Then by Kuratowski 's Theorem -LSB- 5 -RSB- , F contains a subdivsion of a graph H which is isomorphic to K5 or K3 ; 3 . Suppose first that H ? = K5 . Since each v 2 V -LRB- H -RRB- has degree four in H , v 62 I. Hence F < > H unless M = ; which contradicts the minimality of jV -LRB- F -RRB- j since K5 itself has a pair of adjacent mates . So let y 2 V -LRB- G -RRB- n -LRB- V -LRB- H -RRB- -LSB- M -RRB- . G is 3-connected and thus , by the fan lemma -LRB- a wellknown corollary of Menger 's Theorem -LSB- 7 -RSB- -RRB- , there are internally disjoint paths Pyyi , 1 < = i < = 3 , from y to distinct vertices of V -LRB- H -RRB- . It can be seen that each Pyyi may be chosen so as to not contain any pair of mate vertices . With such a choice of the paths Pyyi , ss -LRB- H -RRB- -LSB- 3i = 1 Pyyi is contained in F and contains the expansion of a K3 ; 3 minor of G . So we assume that H ? = K3 ; 3 . By hypothesis , H has two mate vertices of G , x and y that necessarily belong to the same colour class B of H. Note that this implies that the edges ffi -LRB- x -RRB- -LSB- ffi -LRB- y -RRB- of H are short and for every v 2 V -LRB- H -RRB- nB , v 62 I . Now if xy 2 E -LRB- G -RRB- , then neither x nor y belong to I or M and we can include x in I and y in M , leading to a contradiction . If xy 62 E -LRB- G -RRB- , then by our choice of M , one of x , y is in I and we can include the other vertex in M , again a contradiction . It follows that G 2 M. 2 Since T2 has no mate vertices we have the following . Corollary 3.2 The graphs in M are T2-free . Consider the graphs R and Q0 of figure 4 . R is obtained from the cube by adding an edge between two vertices at distance three , and Q0 obtained from R by contracting an edge incident to two vertices of degree three . It is routine to verify the following . 3.2.1 The graph resulting from subdividing any edge of R has T2 as a minor . 3.2.2 If we apply one of the following operations on Q0 , then we obtain a graph that either has T2 as a minor or has R as a subgraph . -LRB- a -RRB- Subdivide an edge of Q0 ? z. -LRB- b -RRB- Subdivide an edge e incident to z and join the newly created vertex with v 2 V -LRB- Q0 -RRB- , v < > v4 when e = zv0 and v < > v5 when e = zv1 . -LRB- c -RRB- Subdivide two edges of Q0 incident to z and join the newly created vertices . QR v v v v v v 1 2 3 4 5 z Figure 4 : The graphs R and Q0 Let Q1 be obtained from Q0 by subdividing with x2 the edge zv1 and by adding the edge x2v5 . Let Q2 be obtained from Q1 by subdividing with x3 the edge zv0 and by adding the edge x3v4 . Note that the graphs R , Q0 , Q1 and Q2 are T2-free but are not mate graphs of planar graphs . Lemma 3.3 Let G be a 3-connected , T2-free graph with at least nine vertices . If G 62 M , then it contains a subgraph L isomorphic to one of R , Q0 , Q1 , Q2 and no proper subdivision of L. Proof . We first show that G contains a subdivision of one of R , Q0 , Q1 , Q2 . By Lemma 3.1 , G has a minor H ? = K3 ; 3 that has no pair of mate vertices of G. Moreover , H has at least one long edge . Indeed , if this is not true , then by the fan lemma , there exist internally disjoint paths Pyyi , 1 < = i < = 3 , from y to distinct vertices of H . Our assumption on the edges of H implies that fy1 ; y2 ; y3g is a colour class B of H and each Pyyi is an edge . Moreover , if y0 is any other vertex of v -LRB- G -RRB- nV -LRB- H -RRB- , then y , y0 and the vertices of H ? B are mutually disjoint . It follows that G 2 M , a contradiction . Label the colour classes of H by fv0 ; v2 ; v4g , fv1 ; v3 ; v5g . Since G is 3-connected and has at least one long edge e , there exists an H-external path Pzz1 of G with z being an internal vertex in the expansion of e . In addition , Pzz1 and e can be selected so that the second attachment edge f of Pzz1 is different from e . If now G does not have T2 as a minor , by symmetry , we may assume that e = v3v2 and f = v3v0 . It is now straightforward to verify that the edges of H ? v3 are short . Moreover , since G is 3-connected and H has no mate vertices of G , there exists an H-external path Pz2v with v = v1 or v5 and z2 62 V -LRB- ss -LRB- H ? v3 -RRB- -RRB- . Without loss of generality , v = v1 . Let H be obtained from ss -LRB- H -RRB- by contracting all edges not incident to a vertex of V -LRB- H -RRB- -LSB- fz ; z1 ; z2g and then adding the edges zz1 , vz2 . Depending on how z1 can be chosen , we distinguish two cases . 1 . z1 < > v0 . If z2 < > z , then one side of the bipartition of the degree three vertices of a T2 minor of G is given by fz ; v1 ; v5g . The other can be selected from fv2 ; v4 ; z2g , fv0 ; v2 ; z2g , fv0 ; v3 ; v5g and fv0 ; v4 ; z2g , respectively , when z2 2 ss -LRB- v0z1 -RRB- -LSB- ss -LRB- zz1 -RRB- , z2 2 ss -LRB- v3 ; v4 -RRB- -LSB- ss -LRB- zv3 -RRB- , z2 2 ss -LRB- z1v3 -RRB- and z2 2 ss -LRB- zv2 -RRB- . Also , when z2 = z , the required bipartition is given by fv0 ; v4 ; zg , fv1 ; v3 ; v5g . 2 . z1 = v0 . Suppose that z2 62 ss -LRB- v3z -RRB- . If z2 is an internal vertex of ss -LRB- v3v4 -RRB- or ss -LRB- v0v3 -RRB- , the bipartition of the degree three vertices of a T2 minor of G is given respectively by fv1 ; v3 ; v5g , fv0 ; v2 ; z2g or fv0 ; v2 ; v3g , fz ; z2 ; v5g . Similarly , if z2 is an internal vertex of ss -LRB- zv0 -RRB- or ss -LRB- zv2 -RRB- , the required bipartition is given respectively by fv2 ; v4 ; z2g , fz ; v1 ; v5g or fv1 ; v5 ; zg , fv0 ; v4 ; z2g . Consider now the possibility of z2 being a vertex of ss -LRB- zv3 -RRB- . If z2 < > z or v3 , then H is a bipartite graph isomorphic to R . If z2 = z or v3 , then H is an isomorphic to Q minor of G. To conclude , consider H and suppose that it is T2-free . If all of its edges are short , then the lemma holds . So assume that e is a long edge of H . By -LRB- 3.2.1 -RRB- and -LRB- 3.2.2 -RRB- , H ? = Q0 and e = zv1 or zv0 , say , e = zv1 . Now G has an H external path whose one end , call it , x2 is an internal vertex of zv1 . By -LRB- 3.2.2 -RRB- , the other must be v5 . Now H + x2v4 ? = Q1 . Moreover , since x2 and v2 are mates in H + x1v5 and all the edges incident to v2 are short , then the edges incident to x2 are also short . Again , if all edges of H + x2v4 are short , the lemma holds . Otherwise , zv0 is the long edge of H + x2x4 . Repetition now of the above argument , with zv0 , yields a minor and subgraph of G which is isomorphic to Q2 , as required . 2 We are now ready to prove the main theorem . Theorem 3.4 Let G be a 3-connected graph , jV -LRB- G -RRB- j > = 9 . G is T2-free if and only if G 2 M -LSB- K. Proof . It is easy to verify that the graphs in K are T2-free . Moreover , by Corollary 3.2 , the graphs in M are T2-free . Thus , if G 2 M -LSB- K , then G is T2-free . Conversely , suppose that G 62 M. By Lemma 3.3 , G has a subgraph L isomorphic to one of R , Q0 , Q1 , Q2 , but not proper subdivision of L . Since G is 3-connected , by the Fan Lemma , there exist internally disjoint paths Pyyi , 1 < = i < = 3 , from y to distinct vertices of L . If L ? = R , then by -LRB- 3.2.1 -RRB- , N -LRB- y -RRB- is a subset of a colour class B of L and each Pyyi is an edge . Moreover , for any vertex y0 of V -LRB- G -RRB- nV -LRB- L -RRB- , N -LRB- y0 -RRB- ? B for otherwise we can obtain a T2-minor whose one colour class contains y0 and two vertices in N -LRB- y -RRB- and the other three vertices in N -LRB- y0 -RRB- . So , we may assume that G does not contain R as a subgraph . If L ? = Q0 , then , by -LRB- 3.2.2 -RRB- , N -LRB- y -RRB- is a subset of a colour class of L ? z . Now if y0 2 V -LRB- G -RRB- n -LRB- V -LRB- L -RRB- -LSB- fyg , then N -LRB- y0 -RRB- = N -LRB- y -RRB- , for otherwise a T2 minor of G can be formed from the bipartition fy ; y0 ; zg , fv0 ; v1 ; v3g . If L ? = Q1 , then again by -LRB- 3.2.2 -RRB- , y can only be a non-adjacent mate of either v2 or v4 and v2 , x2 are non-adjacent . Finally , when L ? = Q2 , y can only be a non-adjacent mate of v2 or v3 . In all cases , G 2 K , as required . 2 A 1-cut reduction of a graph G simply involves splitting of G along a cut vertex . Call a 2-cut S of a graph G proper , if either G?S has at least three components or it has two components each one of them non-trivial . A proper 2-cut reduction involves splitting the graph along a proper 2-cut S and augmenting each non-trivial connected component of the resulting graph by a path of length two joining the vertices of S. Theorem 3.4 allows us to describe in general T2-free graphs . Note first that a graph G is T2-free if and only if each connected component of the graph obtained from a 1 - or a proper 2-cut reduction on G is T2-free . Moreover , mate graphs of planar graphs are T2-free . Thus , a T2-free graph can be reduced by 1 - and proper 2-cuts to a mate graph of a planar graph , or a subdivision of a 3-connected graph in K , or a subdivision of a 3-connected graph with at most eight vertices . We remark that it is possible and relatively easy to determine which subdivisions of K are T2-free . However , their description turns out to be rather technical and we omit it . 4 Mate Reduction and Colouring We now apply the above results to prove the following . Theorem 4.1 If every planar r-graph is r-edge colourable , then every T2-free r-graph is r-edge colourable . Proof . We first consider graphs with a small number of vertices . 4.1.1 Every r-graph G with at most eight vertices is r-edge colourable Proof . If G has an r-edge cut ffi -LRB- S -RRB- such that , jSj , jV -LRB- G -RRB- ? Sj > = 2 and S has odd cardinality , then we can contract respectively S and G ? S to a single vertex and reduce our problem to colouring two graphs with fewer than G vertices . Thus we may assume that G has no such cut . It suffices now to show that that G has a matching M such that G?M is an -LRB- r ? 1 -RRB- - graph . By Tutte 's Theorem -LSB- 13 -RSB- , G has a perfect matching M . We claim that G?M is an -LRB- r ? 1 -RRB- - graph . Indeed , consider any S ae V -LRB- G -RRB- of odd cardinality and note that jffiG -LRB- S -RRB- \ M j < = 3 and jffiG -LRB- S -RRB- j > r imply that jffiG0 -LRB- S -RRB- j > = jffiG -LRB- S -RRB- j ? 3 > = r ? 2 . It follows that jffiG0 -LRB- S -RRB- j > = r ? 1 since jffiG -LRB- S -RRB- j and r have the same parity and jffiG0 -LRB- S -RRB- j and r have different parity . 2 We proceed now with the proof of the theorem . Let G be a T2-free r-graph . If G is planar , then , by hypothesis , it can be coloured as required . Thus we assume that G is not planar and that any graph with fewer vertices than G that meets the hypothesis , is r-edge colourable . We may also assume that G is connected . Suppose first that G is not 3-connected and note that since V -LRB- G -RRB- has even cardinality , G does not have 1-cuts . Thus , G = G1 -LSB- G2 , where V -LRB- G1 -RRB- \ V -LRB- G2 -RRB- consists of exactly two vertices . Since , G is an r-graph , the cardinalities of V -LRB- G1 -RRB- and V -LRB- G2 -RRB- are of the same parity . Moreover , observe that when S : = V -LRB- G1 -RRB- , S0 : = V -LRB- G2 -RRB- and S has odd cardinality , then jffi -LRB- S -RRB- j = r , since jffi -LRB- S -RRB- j + jffi -LRB- S0 -RRB- j > = 2r and jffi -LRB- S -RRB- j + jffi -LRB- S0 -RRB- j < = jffi -LRB- v -RRB- j + jffi -LRB- u -RRB- j = 2r . Similarly , jffi -LRB- S -RRB- j = r when S = V -LRB- G1 -RRB- nfug , S0 = V -LRB- G2 -RRB- nfvg and S has odd cardinality . Thus , when G has a 2-cut one can find a subset of vertices S such that , jSj = 2k + 1 , k > = 1 , and jffi -LRB- S -RRB- j = r. Also , when S and G ? S are contracted to a single vertex , the resulting two graphs are T2-free . It follows now , by induction , that one can r-edge colour the resulting smaller graphs and then combine the solutions found , in the obvious manner , to obtain a colouring of G. So , we may assume that G is 3-connected . Theorem 3.4 tells us that the underlying simple graph of G is either in M or in K . In the latter case , since G is an r-graph , it must have at most eight vertices and by -LRB- 4.1.1 -RRB- , it is r-edge colourable . In the former case we will have to deal with mate vertices . We do this below . Let W : = fw ; w0g be mate vertices of G and let X : = N -LRB- w -RRB- nfw0g . Since G is 3-connected , jXj = 3 and X = N -LRB- w0 -RRB- nfwg . Denote by x1 , x2 , x3 the members of X and let t = jfe : e joins w and w0gj . Let H be the subgraph of G induced by X -LSB- W after the edges with both ends in either X or W are removed . H is a bipartite graph and jffiH -LRB- w -RRB- j = jffiH -LRB- w0 -RRB- j = r ? t. Suppose , momentarily , that r ? t is also the maximum degree of H. Then , it is well known that H is -LRB- r ? t -RRB- - edge colourable . Moreover , every colour class is incident to both vertices of W . Now for every colour class Mk = fwxi ; w0xjg , 1 < = k < = r ? t , we substitute in G an edge eMk joining xi to xj . Let G0 be the graph resulting from G after each colour class of H has been considered and after deleting W . Clearly , G0 is T2-free . Suppose also that G0 is an r-graph . Then we know , from the induction hypothesis , that G0 is r-edge colourable . Given any r-colouring of G0 one can obtain an r-colouring of G as follows . Every edge that belongs to some colour class Mk is assigned the colour of eMk . Every edge that belongs to both G and G0 is simply assigned its colour in G0 . One can now verify that the colouring so obtained is a proper colouring of G ? fe : e joins w and w0g . Moreover , the same r ? t colours are incident to w and w0 . Hence , the remaining t colours can be used to colour fe : e joins w and w0g and thus obtain an r-colouring of G . We need to verify our assumptions regarding G0 and H. First we show that the maximum degree of H is r ? t. Let x be any element of X and let S = W -LSB- fxg . Then , jffiG -LRB- S -RRB- j = 2r ? 2t ? jffiH -LRB- x -RRB- j + r ? jffiH -LRB- x -RRB- j. Moreover , since G is an r-graph , we also have that jffiG -LRB- S -RRB- j > = r . It follows that jffiH -LRB- x -RRB- j < = r ? t , as required . To complete the proof of the theorem we need to establish that G0 is an r-graph . Let S be any odd cardinality subset of V -LRB- G0 -RRB- . If jS \ Xj = or 1 , then jffiG0 -LRB- S -RRB- j > = r , since jffiG0 -LRB- S -RRB- j = jffiG -LRB- S -RRB- j . If jS \ Xj = 3 , then again jffiG0 -LRB- S -RRB- j > = r , since jffiG0 -LRB- S -RRB- j = jffiG -LRB- S -LSB- W -RRB- j. Thus , we may assume that jS \ Xj = 2 . Let fxg = Xnfxi ; xjg and observe that in G0 , exactly jffiH -LRB- x -RRB- j edges of the form eMk have one end in x and the other in fxi ; xjg . Therefore , jffiG0 -LRB- S -RRB- j = jffiG -LRB- S -LSB- W -RRB- j?jffiH -LRB- x -RRB- j + jffiH -LRB- x -RRB- j = jffiG -LRB- S -LSB- W -RRB- j > = r , as required . 2 Acknowledgements : The authors respectively wish to thank Universit?e du Queb?ec ? a Montr?eal and the Center for Discrete Mathematics and Computer Science for their support and hospitality . References -LSB- 1 -RSB- S. Baum and L.E. Trotter Jr. . Integer rounding for polymatroid and branching optimization problems . SIAM journal on Algebraic and Discrete Methods , 2:416 -LCB- 425 , 1981 . -LSB- 2 -RSB- J. Edmonds . Maximum matching and the polyhedron with 0 , 1 verices . J. of Research of the National Bureau of Standards -LRB- B -RRB- , 69:125 -LCB- 130 , 1965 . -LSB- 3 -RSB- A.K. Kelmans . 3-connected graphs without essential 3-cuts and triangles . Nauka Sibirisk Otdel Novosibirisk , 3:513 -LCB- 538 , 1986 . -LSB- 4 -RSB- K. Kilakos and B. Shepherd . Excluding minors in 3-regular graphs . Technical Report 92748 , Institut f?ur ? Okonometrie und Operations Research , Bonn , Germany , 1992 . -LSB- 5 -RSB- C. Kuratowski . Sur le probl?eme des courbes gauches en topologie . Fund . Math. , 15:271 -LCB- 283 , 1986 . -LSB- 6 -RSB- O. Marcotte . On the chromatic index of multigraphs and a conjecture of Seymour -LRB- II -RRB- . In Proceedings of the DIMACS Workshop in Polyhedral Cominatorics , volume 1 . William Cook and Paul D. Seymour eds. , 1990 . -LSB- 7 -RSB- K. Menger . Zur allgemeinen kurrentheorie . Fund . Math. , 10:96 -LCB- 115 , 1927 . -LSB- 8 -RSB- N. Robertson . Personal communication . -LSB- 9 -RSB- P.D. Seymour . On multi-colourings of cubic graphs and conjectures of Fulkerson and Tutte . J. London Math . Society , 38:423 -LCB- 451 , 1979 . -LSB- 10 -RSB- P.D. Seymour . On Tutte 's extension of the four-colour problem . J. Combin . Theory -LRB- B -RRB- , 31:82 -LCB- 94 , 1981 . -LSB- 11 -RSB- P.D. Seymour . Colouring series-parallel graphs . Combinatorica , 10:379 -LCB- 392 , 1990 . -LSB- 12 -RSB- P.G. Tait . Remarks on the colouring of maps . Proc . Royal Soc . Edinburgh Ser . A , 10:729 , 1880 . -LSB- 13 -RSB- W.T. Tutte . The factorization of linear graphs . J. London Math . Soc. , 22:107 -LCB- 111 , 1947 . -LSB- 14 -RSB- W.T. Tutte . On the algebraic theory of graph colourings . J. Combin . Theory -LRB- B -RRB- , 23:15 -LCB- 50 , 1966 .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

hilbert basis
integer decomposition property
minor
petersen graph