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The concept of the $k$-pairable graphs was introduced by Zhibo Chen (On $k$-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter $p\left(G\right)$, called the pair length of a graph $G$, as the maximum $k$ such that $G$ is $k$-pairable and $p\left(G\right)=0$ if $G$ is not $k$-pairable for any positive integer $k$. In this paper, we answer the two open questions raised by Chen in the case that the graphs involved...

We give a necessary and sufficient condition for the existence of perfect matchings in a plane bipartite graph in terms of elementary edge-cut, which extends the result for the existence of perfect matchings in a hexagonal system given in the paper of F. Zhang, R. Chen and X. Guo (1985).