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Keywords:
degree sequence; bigraphic pair; potentially ${\rm SB}_{s+s',t+t'}$-bigraphic pair
Summary:
Let $K_{s,t}$ be the complete bipartite graph with partite sets $\{x_1,\ldots ,x_s\}$ and $\{y_1,\ldots ,y_t\}$. A split bipartite-graph on $(s+s')+(t+t')$ vertices, denoted by ${\rm SB}_{s+s',t+t'}$, is the graph obtained from $K_{s,t}$ by adding $s'+t'$ new vertices $x_{s+1},\ldots ,x_{s+s'}$, $y_{t+1},\ldots ,y_{t+t'}$ such that each of $x_{s+1},\ldots ,x_{s+s'}$ is adjacent to each of $y_1,\ldots ,y_t$ and each of $y_{t+1},\ldots ,y_{t+t'}$ is adjacent to each of $x_1,\ldots ,x_s$. Let $A$ and $B$ be nonincreasing lists of nonnegative integers, having lengths $m$ and $n$, respectively. The pair $(A;B)$ is potentially ${\rm SB}_{s+s',t+t'}$-bigraphic if there is a simple bipartite graph containing ${\rm SB}_{s+s',t+t'}$ (with $s+s'$ vertices $x_1,\ldots ,x_{s+s'}$ in the part of size $m$ and $t+t'$ vertices $y_1,\ldots ,y_{t+t'}$ 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 ${\rm SB}_{s+s',t+t'}$-bigraphic. A simplification of this characterization is also presented.
References:
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