On cliques and bicliques

For integers m, n ≥ 2, let g(m, n) be the minimum order of a graph, where every vertex belongs to both a clique K_m of order m and a biclique K(n, n). We show that g(m, n) = 2(m + n − 2) if m ≤ n − 2. Furthermore, for m ≥ n − 1, we establish that ≡ 0 mod(n − 1) or, if m is sufficiently large and is not an integer. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 60–66, 2000