Cayley partitionable graphs

Cayley partitionable graphs

Cayley partitionable graphs A. P^ echer a a LIFO - Universite d'Orleans BP 6759 45067 ORLEANS Cedex 2 - France Abstract In this paper we investi...

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Cayley partitionable graphs A. P^ echer

a

a LIFO

- Universite d'Orleans BP 6759 45067 ORLEANS Cedex 2 - France

Abstract In this paper we investigate the class of Cayley partitionable graphs. This investigation is motivated by the Strong Perfect Graph Conjecture. Cayley partitionable graphs are Cayley Graphs which are closely related to near-factorizations of nite groups. We prove some structural properties of near-factorizations and give examples of Cayley partitionable graphs which are not generated by all constructions of partitionable graphs known so far. Key words: partitionable, perfect, Cayley, graph, near-factorization, group

1 Introduction In 1960, Claude Berge introduced the notion of perfect graph and conjectured that every minimal imperfect graph (i.e. an imperfect graph such that all its induced subgraphs are perfect) is an odd hole or the complement of an odd hole. This conjecture is known as the Strong Perfect Graph Conjecture and is still open.

G is said to be partitionable if there exist two integers p and q such that G has pq + 1 vertices and for every vertex v of G, the induced subgraph G n fv g admits a partition in p cliques of cardinality q and also admits a partition in q stable sets of cardinality p. Following the paper of Bland, Huang and Trotter [2], a graph

Padberg [7] proved that every minimal imperfect graph is partitionable. Thus a counter-example to the Strong Perfect Graph Conjecture would lie in the class of partitionable graphs.

Preprint submitted to Elsevier Preprint

16 May 2000

In 1979, Chvatal, Graham, Perold and Whitesides introduced two constructions for making partitionable graphs [5]. Due to the names of these four authors, we call C GP W1 -graphs the graphs produced by the rst method and C GP W2 -graphs the graphs produced by the second one. There are no counterexamples to the Strong Perfect Graph Conjecture in these two classes [6] [9] [1]. Boros, Gurvich and Hougardy [3] described last year a recursive generation of partitionable graphs extending the rst method of Chvatal, Graham, Perold and Whitesides. We call BGH -graphs the graphs generated by this new construction. It is unknown whether there is a counter-example to the Strong Perfect Graph Conjecture in this wider class. Our work focused on the second construction of Chvatal, Graham, Perold and Whitesides. Before describing the track followed, we rst need to recall some de nitions and some results. A normalized graph is a graph such that for every edge maximum clique containing both i and j .

f g, there exists a i; j

Let G be a nite group of order n with operation . Let S be a symmetric subset of G which does not contain the identity element e. The Cayley graph with connection set S is the simple graph with vertex set G and edge set ffi; j g; i 1  j 2 S g. It was noticed that every C GP W2 -graph is a normalized Cayley partitionable graph of a cyclic group [1]. Thus the class of Cayley partitionable graphs of nite groups is a natural extension of the class of the C GP W2 graphs. Cayley partitionable graphs are closely related with the group theoretic notion of near-factorization as explained below. Two subsets

and B of G of cardinality at least 2 are said to form a nearfactorization of G if and only if n = jAj  jB j + 1 and there is an element u(A; B ) of G such that A  B = G n fu(A; B )g. A

If (A; B ) is a near-factorization of a nite group then the Cayley graph G(A; B ) with connection set (A 1  A) n feg is a normalized partitionable graph [8]. Conversely, if is any Cayley partitionable graph on a group G, then there exists a near-factorization (A; B ) of G such that G(A; B ) is the normalized graph of . This equivalence motivated this paper: we wanted to produce near-factorizations of some nite groups, so as giving rise to 'new' partitionable graphs. At rst, things seem to go pretty well: beyond the class of the cyclic groups, 2

all the dihedral groups have near-factorizations. In this paper, we give a basic result explaining the relation between near-factorizations of the cyclic groups and near-factorizations of the dihedral groups, for half of the dihedral groups. Furthermore, all C GP W2 -graphs of even order arise from near-factorizations of dihedral groups [8]. Unfortunately, we did not succeed in nding out a nearfactorization of a dihedral group, whose associated partitionable graph is not a C GP W2 -graph. Thus we needed to study the near-factorizations of other groups. In the abelian case, it is known that such a group would be of order at least 92 [4].

2 Main result The main result of our work is Theorem 1 below which states that every nearfactorization (A; B ) of a given group G splits equally in the cosets of any normal subgroup of index at most 4, informally speaking.

Theorem 1 Let H be any normal subgroup of G of index d with (A; B ) is a near-factorization of G then for every g in G, we have

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j j j \ A

A

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and

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j j j \ B

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j j j

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j j j B d

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. If

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Theorem 1 is useful to prove that some groups do not have any near-factorizations at all. It may be used to speed up computer experiments. In particular, it helped us to nd out near-factorizations in groups which are not cyclic, nor dihedral. The associated Cayley partitionable graphs turn out not to be BGH graphs or C GP W2 -graphs.

References [1] G. Bacso, E. Boros, V. Gurvich, F. Ma ray, and M. Preissmann, On minimally imperfect graphs with circular symmetry, J. Graph Theory 29 (1998), 209{225. [2] R.G. Bland, H.C. Huang, and L.E. Trotter, Graphical minimal imperfection, Disc. Math. 27 (1979), 11{22.

properties related to

[3] E. Boros, V. Gurvich and S. Hougardy, Recursive generation of partitionable graphs, Tech. Report RR 10-99, Rutcor Research Report, June 1999.

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[4] D. De Caen, D.A. Gregory, I.G. Hughes, and D.L. Kreher, Near-factors of nite groups, Ars Combinatoria 29 (1990), 53{63. [5] V. Chvatal, R.L. Graham, A.F. Perold, and S.H. Whitesides, Combinatorial designs related to the perfect graph conjecture, Annals Discrete Math. 21 (1984), 197{206. [6] C.M. Grinstead, On circular critical graphs, Disc. Math. 51 (1984), 11{24. [7] M.W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974), 180{ 196. [8] A. P^echer, Graphes partitionnables associes aux quasi-factorisations des groupes nis, Research Report RR-LIFO-2000-06, LIFO, Laboratoire d'Informatique Fondamentale d'Orleans, Universite d'Orleans, BP 6759, F45067 Orleans Cedex 2, 2000. [9] A. Seb o, On critical edges in minimal imperfect graphs, J. Comb. Theory Series B 67 (1996), 62{85.

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