Graphical properties related to minimal imperfection

Abstract

Say that graph G is partitionable if there exist integers α⩾2, ω⩾ 2, such that | V( G)| ≡ αω + 1 and for every υ ϵ V( G) there exist partitions of V( G)\\ υ into stable sets of size α and into eliques of size ω. An immediate consequence of Lovász' characterization of perfect graphs is that every minimal imperfect graph G is partitionable with α ≡ α ( G) and ω ≡ ω( G). Padberg has shown that in every minimal imperfect graph G the cliques and stable sets of maximum size satisfy a series of conditions that reflect extraordinary symmetry G. Among these conditions are: the number of cliques of size ω( G) is exactly |V( G)|; the number of stable sets of size α( G) is exactly |V( G)|: every vertex of G is contained in exactly ω( G) cliques of size ω( G) and α( G) stable sets of size α( G): for every clique Q (respectively, stable set S) of maximum size there is a unique stable set S (clique O) of maximum size such that Q∩ S ≡ Ø. Let C n k denote the graph whose vertices can be enumerated as υ 1,…,υ n in such a way that υ 1 and υ 1 are adjacent in G if and only if i and j differ by at most k, modulo n. Chvátal has shown that Berge's Strong Perfect graph Conjecture is equivalent to the conjecture that if G is minimal imperfect with α( G) ≡ α and ω( G) ≡ ω, then G has a spanning subgraph isomorphic to C αω+1 ω. Padberg's conditions are sufficiently restrictive to suggest the possibility of establishing the Strong Perfect Graph Conjecture by proving that any graph G satisfying these conditions must contain a spanning subgraph isomorphic to C αω+1 ω , where α( G) ≡ α and ω( G) ≡ ω. It is shown here, using only elementary linear algebra, that all partitionable graphs satisfy Padberg's conditions, as well as additional properties of the same spirit. Then examples are provided of partitionable graphs which contain no spanning subgraph isomorphic to C αω+1 ω , where α( G) ≡ α and ω( G) ≡ ω.