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

TL;DR

This paper investigates the Helly property in complements of regular graphs, establishing conditions under which these complements are not Helly and exploring the relationship between Hellyness and clique graph convergence.

Contribution

It provides new conditions for non-Helly complements of regular graphs and examines the equivalence of Hellyness and clique convergence properties.

Findings

Complement of a k-regular graph with sufficiently large n is not Helly.

Conditions under which Hellyness does not hold for graph complements.

Exploration of the relationship between Hellyness and clique graph convergence.

Abstract

A collection of sets is intersecting, if any pair of sets in the collection has nonempty intersection. A collection of sets \(\mathcal{C}\) 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 \(n>3k+\sqrt{2k^{2}-k}\), then the complement \(\bar{G}\) 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 \(k\).