Discrete 32 (1980) 331-334 (~) North-Hol!and Publishing Company

COMMUNICATION

EVEN POLYHEDRAL DECOMPOSITIONS OF CUBIC GRAIPHS

M. PREISSMAN ~ I.M.A.G.. B.P. 53X, 78041 Grenoble C(:dex, France

Communicated by J. C. Bermond Received June lt,~8l)

An even polyhedral decomposition of a finite cubic grap;'i G is defined a-, a sel of elem,:nlar~ cycles of even length ir~ G with the property that each edge of G lies in exactly two of them. l~" G has chromatic index three, then G has an e~en !polyhedral decomposition. We ~d~ow ~hat. contrary to a theorem of Szekeres [2]. this property (m have an even p¢~lyhcdra! decompo~iliom doesn't characterize the cubic graphs of cttromatic inde~ three. In particuizm there exit,Is mq infinite family of sharks all having an even polyhedJal decomposiiion.

Introduction

All graphs considered here are finite, undirected and without loops: the reader is referred to [1] for general definitions.

Definition. We ,':_all even polyhedral decomposition (Briefly E.P decomposition) of a cubic graph G a set D of elementary cycles c,f even length such ~hat every edge of G belongs to exactly two cycles of D.

Szekeres [2] remarked that if t~ = (X, E) is a cubic graph ,of chromatic index three, then G has an E.P. decomposition. Indeed, given any particular three-coloring of the edges, it is easy to check thai the se' of the two-coloured elementary cycles is an E.P. decomposition. Szekeres has befieved to show that the converse result aiso turns out to be true, that is to say that for any E.P. decomposition D of a cubic graph G, there exists an edge three-colouring of G such that ~11 cycles of D are two-coloured. This la~t assertion is false. In,.teed, let us consider the graph of Fig. 1 together with an E.P. decomposition D consisling of five cycles labelled I, 2, 3, 4, 5. Suppose that there exists a three-colouring of the edges of this graph, such that a2 cycles of D ~re two- coloured. Then "',e have: (1) the cycles and 2 intersect in two edges A and B. Thes have therefore exactly one colour in common, and A and B must be coloured with this colour. 331 332 M. ?~eissmann

45

5 1

2a 24

34

Fig. 1.

(2) in the cycle l, the edges A and B are separated by an even number of edges, thereft,re they must bear two distinct colours. We come to a contradiction. But we observe that the graph of Fig. 1 is edge three-colorable. Thus. we might think that the weak version of Szekeres's theorem given by Fiorini and Wilson [3]: any cubic graph having an E.P. decomposition has chromatic index three, is true. This is not the case; indeed in the following paragraph we will give examples of cubic graphs having an E.P. decomposition and nevertheless of chromatic index four. On the other hand, J. C. Fourniei [4] remarked that if Szekeres's theorem was true then, using a theorem of Fisk [5], one can obtain a very short proof for the four colour theorem.

Snarks mJd counter examples

1, Definition In 1975, lsaacs [6] defined what he called a non trivial cubic graph of chromatic index four~ and which was ',o,ter christened by Martin Gar(ner [7]: it is a cubic graph of chlomatic index four, cyclically edge 4-connected. The first and the most famous snark known l~ the Peter,~;en graph. It is a!so the smallest. Fig. 2 ~tfers two drawings of this graph. It is easy to show that the has no E.P. decomposition and so c~nnot give a countel example.

Fig. 2. Even ~lyhedral decompositions o[ cubi~ graphs 333 V: Fig. 3. Y. Fig, 4. J~ (k =2f~ 3).

II. The flower-~narks This is an infinite family of sharks discovered by Isaacs [6]. They are built starting from the graph Y with six pendants (three extending to the left ::nd ti~ree to the right) shown in Fig. 3. The study of the permissible edge 3-colduring:~ of these pendants allows to show that a cyclic comp :-und of an odd number k of Y (described in Ng. 4) is a graph Jk which cannot be edge 3-coloured. J3 is not a snark since it contains a triangle. If we replace this triangle by a single , we obt~ir, the Petersen graph. The Jk (k >3, k odd) ~re also called flower-snarks~

IIl. Even polyhedral decompositions of flower snc:rks J3 cannot have an E.P. decomposition, since the Pc'~ersen graph doesn't have one. On the other hand, there exists one for J5 which is exhibited in Fig. 5. Now, we remark that we can delete the three edges of J~ marked A, B, C and replace them by the graph of Fig. 6 at the indicated vertices. U.A. Celmins in [9] uses this operation, and calls it a splicing.

3 4 Fig. 5. 334 M. tS"ei'c, mann

2 .1 t 2 2 3

-, 1 2 lla/ . 2 3 lla/ 1 2 , ~2

Fig. 6.

The: edges numbered 1 (2, 3) still form an elementary cycle of even length. Each edge lies on exactly two cycles, ttenceo we have an E.P. deco.,nposition of Jr. By repeating this operation, we obtain an E.P. decomp)sition with six cycles, for any .Ik (k>3./¢ odd).

Remark, Using constructions due to '~. ,cs [6] (see also [8]) it is possible to build cubic graphs of chromatic index !ou, with edge-connectivity two. three or cyclic- edge-connectivity four. having E.P. decompositions. The "double star snark" of Isaac.~ has one too.

W,~: conclude by noting that bridgeless cubic graphs which doesn't ha'~e an E.P. decomposition seem to be rather scarce.

Refe~ences

[ 1J C. Berge. Graphs and Hypergr:~ph~,. North-Itolt[~nd Mathematical Library. Vol. 6 (Norlh-Holland. Amsterdam. 19761, [2] C,. Szekeres. Polyhedral decompositions of cubic graphs. Bull. Aust. Math. 3oc. 8 ( 19731 3t~7.387. [3] S. Fiori:~i and R. J. Wilson. E,'tge-colourings of graphs. Research Notes in Mathematics (Pi:man. I (,~77) 5~ [41 ~. C. Fournier. ('ommunica.iol~ er sdminail'e ,5 la Maison des Sciences de l'Homme rPari,,. ~'74-75L [SJ Steve Fisk. ('omhinatorial structures on triangulati, ms II. Local colorings, Adv. in Math. ! I (! 973) th. 1(). p. 349. [,~] R. Isaacs. Infinite familie'~ of non-trivial trivalem graphs which are not Tait-colourable. Amer. Math Momhb. g2 (31 (1975! 221-239. [71 M. Garry. .Mathematicai Games. Scientific ,~maer. 234 (1976) 126-130. [ i U A. ('ehnins. J. L. Fouqucl and E. R. SwaT'~. Consmrc,fion and characterization of s,mrks. J. (to appeart. [9] U. A. Celmins, Study of three conjectures on an infinite family of snarks. Research Rcport COItR 79-19. Uni:ersity of Waterloo.