An extension of a result of Alon, Ben-Shimon and Krivelevich on bipartite graph vertex sequences

Let S=(a1,…,am;b1,…,bn) be a pair of two nonincreasing sequences of positive integers, that is, a1,…,am and b1,…,bn are two nonincreasing sequences of positive integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. In this paper, we give a simple sufficient condition for a pair S=(a1,…,am;b1,…,bn) to be a bigraphic pair. The condition depends only on the lengths of two sequences of the pair and their largest and smallest elements. This result extends a result due to Alon et al. (2010).