A note on graphs with a prescribed adjacency property

Abstract

Let m and n be nonnegative integers and k be a positive integer. A graph G is said to have property P(m, n, k) if for any set of m + n distinct vertices of G there are at least k other vertices, each of which is adjacent to the first m vertices of the set but not adjacent to any of the latter n vertices. The problem that arises is that of characterising graphs having property P(m, n, k). This problem has been considered by several authors and a number of results have been obtained. In this paper, we establish a lower bound on the order of a graph having property P(m, n, k). Further, we show that all sufficiently large Paley graphs satisfy properties P(1, n, k) and P(n, 1, k).