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

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.