The EKR-module property of pseudo-Paley graphs of square order

Abstract

We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erdős-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combination of characteristic vectors of canonical cliques. This extends the EKR-module property of Paley graphs of square order and solves a problem proposed by Godsil and Meagher. Different from previous works, which heavily rely on tools from number theory, our approach is purely combinatorial in nature. The main strategy is to view these graphs as block graphs of orthogonal arrays, which is of independent interest. Theorem 1.1 ([Blo84]). Let q be an odd prime power. The Paley graph P q 2 satisfies the strict-EKR property.Godsil and Meagher [GM16, Section 5.9] call Theorem 1.1 the EKR theorem for Paley graphs. Theorem 1.1 was first proved by Blokhuis [Blo84]. Extensions and generalizations of Theorem 1.1