ABOUT AN EXTREMAL PROBLEM OF BIGRAPHIC PAIRS WITH A REALIZATION CONTAINING K_s,t

Let p = (f(1), ... , f(m);g(1), ... , g(n)), where f(1), ... , f(m) and g(1), ... , g(n) are two non-increasing sequences of nonnegative integers. The pair p = (f(1), ... , f(m); g(1),..., g(n)) is said to be a bigraphic pair if there is a simple bipartite graph G = (X U Y, E) such that f(1), ... , f(m )and g(1), ..., g(n) are the degrees of the vertices in X and Y, respectively. In this case, G is referred to as a realization of p. We say that p is a potentially K-s,K-t-bigraphic pair if some realization of p contains K-s,K-t(with s vertices in the part of size m and t in the part of size Theory 29 (2009) 583-596] defined s(K-s,K-t, m, n) to be the minimum integer k such that every bigraphic pair p = (f(1), ... , f(m); g(1), ... , g(n)) with s(p) = for n = m = 9s(4)t(4). In this paper, we first give a procedure and two sufficient conditions to determine if p is a potentially K-s,K-t-bigraphic pair. Then, we determine s(K-s,K-t, m, n) for n = m = s and n = (s + 1)t(2) - (2s - 1)t + s - 1. This provides a solution to a problem due to Ferrara et al.