English

Abstract

Let Ks,t be the complete bipartite graph with partite sets {x1, …, xs} and {y1, …, yt}. A split bipartite-graph on (s + s′) + (t + t′) vertices, denoted by SBs + s′, t + t′, is the graph obtained from Ks,t by adding s′ + t′ new vertices xs + 1, …, xs + s′}, yt + 1, …, yt + t′ such that each of xs + 1, …, xs + s′; is adjacent to each of y1, …, yt and each of yt + 1, …, yt + t′ is adjacent to each of x1, …, xs. Let A and B be nonincreasing lists of nonnegative integers, having lengths m and n, respectively. The pair (A; B) is potentially SBs + s′, t + t′-bigraphic if there is a simple bipartite graph containing SBs + s′, t + t′ (with s + s′ vertices x1, …, xs + s′ in the part of size m and t + t′ vertices y1, …, yt + t′ in the part of size n) such that the lists of vertex degrees in the two partite sets are A and B. In this paper, we give a characterization for (A; B) to be potentially SBs + s′, t + t′-bigraphic. A simplification of this characterization is also presented.