About an extremal problem of bigraphic pairs with a realization containing Ks,t

Let π = (f1, . . . , fm; g1, . . . , gn), where f1, . . . , fm and g1, . . . , gn are two non-increasing sequences of nonnegative integers. The pair π = (f1, . . . , fm; g1, . . . , gn) is said to be a bigraphic pair if there is a simple bipartite graph G = (X ∪ Y,E) such that f1, . . . , fm and g1, . . . , gn are the degrees of the vertices in X and Y , respectively. In this case, G is referred to as a realization of π. We say that π is a potentially Ks,t-bigraphic pair if some realization of π contains Ks,t (with s vertices in the part of size m and t in the part of size n). Ferrara et al. [Potentially H-bigraphic sequences, Discuss. Math. Graph Theory 29 (2009) 583–596] defined σ(Ks,t,m, n) to be the minimum integer k such that every bigraphic pair π = (f1, . . . , fm; g1, . . . , gn) with σ(π) = f1+· · ·+fm ≥ k is potentiallyKs,t-bigraphic. They determined σ(Ks,t,m, n) for n ≥ m ≥ 9st. In this paper, we first give a procedure and two sufficient conditions to determine if π is a potentially Ks,t-bigraphic pair. Then, we determine σ(Ks,t,m, n) for n ≥ m ≥ s and n ≥ (s+ 1)t− (2s− 1)t+ s− 1. This provides a solution to a problem due to Ferrara et al.