Potentially H-bigraphic sequences

We extend the notion of a potentially H-graphic sequence as follows. Let A and B be nonnegative integer sequences. The sequence pair S = (A,B) is said to be bigraphic if there is some bipartite graph G = (X ∪ Y,E) such that A and B are the degrees of the vertices in X and Y , respectively. If S is a bigraphic pair, let σ(S) denote the sum of the terms in A. Given a bigraphic pair S, and a fixed bipartite graph H, we say that S is potentially H-bigraphic if there is some realization of S containing H as a subgraph. We define σ(H, m, n) to be the minimum integer k such that every bigraphic pair S = (A,B) with ΙAΙ = m, ΙBΙ = n and σ(S) ≥ k is potentially H-bigraphic. In this paper, we determine σ(K_s,t, m,n); σ(P_t; m, n) and σ(C_2t, m, n).