A note on graphs with disjoint maximal cliques ans a link with evasiveness

TL;DR

This paper proves that finite vertex-transitive graphs with a clique intersecting all maximal cliques are complete, providing insights into the evasiveness conjecture and answering questions posed by Berge and Payan.

Contribution

It establishes a new characterization of vertex-transitive graphs with certain clique properties, advancing understanding of the evasiveness conjecture.

Findings

Vertex-transitive graphs with intersecting maximal cliques are complete.

Provides a positive answer to a case of the evasiveness conjecture.

Addresses a question raised by Berge and Payan.

Abstract

In this note, we prove that a finite vertex-transitive graph which has a clique which intersects all maximal cliques is a complete graph. This gives a positive answer in the case of vertex-transitive graphs to a question raised by Berge and Payan. It also gives a positive answer to a special case of the evasiveness conjecture.