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 g(m, n) = [(√m - 1 + √n - 1)²] if [√(m - 1)(n - 1)] ≡ 0 mod(n - 1) or, if m is sufficiently large and √(m - 1)(n - 1) is not an integer.