On the clique behavior and Hellyness of the complements of regular graphs

Abstract

A collection of sets is intersecting, if any pair of sets in the collection has nonempty intersection. A collection of sets has the Helly property if any intersecting subcollection has nonempty intersection. A graph G is Helly if the collection of maximal complete subgraphs of G has the Helly property. We prove that if G is a k-regular graph with n vertices such that n>3k+2k2−k, then the complement G‾ is not Helly. We also consider the problem of whether the properties of Hellyness and convergence under the clique graph operator are equivalent for the complement of k-regular graphs, for small values of k.