# Bigraphic pairs with a realization containing a split bipartite-graph

• Volume: 69, Issue: 3, page 609-619
• ISSN: 0011-4642

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## Abstract

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Let ${K}_{s,t}$ be the complete bipartite graph with partite sets $\left\{{x}_{1},...,{x}_{s}\right\}$ and $\left\{{y}_{1},...,{y}_{t}\right\}$. A split bipartite-graph on $\left(s+{s}^{\text{'}}\right)+\left(t+{t}^{\text{'}}\right)$ vertices, denoted by ${\mathrm{SB}}_{s+{s}^{\text{'}},t+{t}^{\text{'}}}$, is the graph obtained from ${K}_{s,t}$ by adding ${s}^{\text{'}}+{t}^{\text{'}}$ new vertices ${x}_{s+1},...,{x}_{s+{s}^{\text{'}}}$, ${y}_{t+1},...,{y}_{t+{t}^{\text{'}}}$ such that each of ${x}_{s+1},...,{x}_{s+{s}^{\text{'}}}$ is adjacent to each of ${y}_{1},...,{y}_{t}$ and each of ${y}_{t+1},...,{y}_{t+{t}^{\text{'}}}$ is adjacent to each of ${x}_{1},...,{x}_{s}$. Let $A$ and $B$ be nonincreasing lists of nonnegative integers, having lengths $m$ and $n$, respectively. The pair $\left(A;B\right)$ is potentially ${\mathrm{SB}}_{s+{s}^{\text{'}},t+{t}^{\text{'}}}$-bigraphic if there is a simple bipartite graph containing ${\mathrm{SB}}_{s+{s}^{\text{'}},t+{t}^{\text{'}}}$ (with $s+{s}^{\text{'}}$ vertices ${x}_{1},...,{x}_{s+{s}^{\text{'}}}$ in the part of size $m$ and $t+{t}^{\text{'}}$ vertices ${y}_{1},...,{y}_{t+{t}^{\text{'}}}$ 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 $\left(A;B\right)$ to be potentially ${\mathrm{SB}}_{s+{s}^{\text{'}},t+{t}^{\text{'}}}$-bigraphic. A simplification of this characterization is also presented.

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