On cliques and bicliques

For integers m, n greater than or equal to 2, let g(m, n) be the minimum or der 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(ml n) = 2(m + n - 2) if m I n - 2. F urthermore, for m greater than or equal to n - I, we establish that g(m, n) = [(root m - 1 + root n - 1)(2)] if [root(m - 1)(n - 1)] = 0 mod(n - I) or , if m is sufficiently large and root(m - 1)(n - 1) is not an integer. (C) 2000 John Wiley & Sons, Inc.