Bigraphic Pairs with a Realization Containing a Split Bipartite-Graph

Abstract

Let K_s,t be the complete bipartite graph with partite sets {x₁, …, x_s} and {y₁, …, y_t}. A split bipartite-graph on (s + s′) + (t + t′) vertices, denoted by SB_{s + s′, t + t′}, is the graph obtained from K_s,t by adding s′ + t′ new vertices x_{s + 1}, …, x_{s + s}′}, y_{t + 1}, …, y_{t + t′} such that each of x_{s + 1}, …, x_{s + s′}; is adjacent to each of y₁, …, y_t and each of y_{t + 1}, …, y_{t + t′} is adjacent to each of x₁, …, x_s. Let A and B be nonincreasing lists of nonnegative integers, having lengths m and n, respectively. The pair (A; B) is potentially SB_{s + s′, t + t′}-bigraphic if there is a simple bipartite graph containing SB_{s + s′, t + t′} (with s + s′ vertices x₁, …, x_{s + s′} in the part of size m and t + t′ vertices y₁, …, y_{t + 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 SB_{s + s′, t + t′}-bigraphic. A simplification of this characterization is also presented.