# Institute of Mathematics

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

#### Jian-Hua Yin, Jia-Yun Li, Jin-Zhi Du, Hai-Yan Li

###### Received February 23, 2017.   Published online May 7, 2019.

Abstract:  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.
Keywords:  degree sequence; bigraphic pair; potentially ${\rm SB}_{s+s',t+t'}$-bigraphic pair
Classification MSC:  05C07
DOI:  10.21136/CMJ.2019.0076-17

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Affiliations:   Jian-Hua Yin, Jia-Yun Li, Jin-Zhi Du, Hai-Yan Li (corresponding author), School of Sciences, Hainan University, Renmin Rd, Meilan Qu, Haikou 570228, P. R. China, e-mail: lhy9694@163.com; yinjh@hainanu.edu.cn

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