A note on cliques and independent sets

For integers m, n ≥ 2, let f(m, n) be the minimum order of a graph where every vertex is in both a clique of cardinality m and an independent set of cardinality n. We show that f(m, n) = d( √ m − 1 + √ n − 1)2e. In this paper we use the terminology of [1]. Specifically, graph G has vertex set V (G), edge set E(G) and order p = |V (G)|. Also, N(v) denotes the neighborhood of a vertex v, i.e., the set of all vertices adjacent to v. Further, an m-clique is a complete subgraph of order m. For positive integers m and n, let f(m,n) be the minimum order of a graph where every vertex is in a clique of cardinality m, and every vertex is in an independent set of cardinality n. All correspondence to this author