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 Km 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 $g(m, n) = \lceil (\sqrt{m - 1} + \sqrt{n - 1})^{2}\rceil $ if $\lceil \sqrt{(m - 1)(n - 1)}\rceil$ ≡ 0 mod(n - 1) or, if m is sufficiently large and $\sqrt{(m - 1)(n - 1)}$ is not an integer. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 60–66, 2000