Maximal cliques in the Paley graph of square order

Abstract

Determining the clique number of the Paley graph of order q, q ≡ 1 (mod 4) a prime power, is a difficult problem. However, the work of Blokhuis implies that in the Paley graph of order q 2, where q is any odd prime power, the clique number is in fact q. In this paper we construct maximal cliques of size 1 2 (q + 1) or 1 2 (q + 3) , accordingly as q ≡ 1 (mod 4) or q ≡ 3 (mod 4), in the Paley graph of order q 2. It is believed that these are the largest maximal cliques which are not maximum. We also briefly discuss maximal cliques in some graphs naturally associated with the interior and exterior points of a conic in PG(2, q) for odd prime powers q.