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 C has the Helly property if any intersecting subcollection has nonempty intersection. A graph 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 + √ 2k 2 − 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.