Bipartite binding number, k-factor and spectral radius of bipartite graphs

Abstract

The binding number $b\left(G\right)$ of a graph G is the minimum value of $\left|N_G\right(S\left)\right|/\left|S\right|$ taken over all non-empty subsets S of $V\left(G\right)$ such that $N_G\left(S\right)\ne V\left(G\right)$. The bipartite binding number $b^{\prime }\left(G\right)$ of a bipartite graph $G=(X,Y)$ is defined to be $\min \left\{\right|X|,|Y\left|\right\}$ if $G=K_{\left|X\right|,\left|Y\right|}$ and$\min \{\underset{\begin{array}{c}\varnothing \ne S\subseteq X\\ N_G\left(S\right)⊊Y\end{array}}{\min }\frac{\left|N_G\right(S\left)\right|}{\left|S\right|},\underset{\begin{array}{c}\varnothing \ne T\subseteq Y\\ N_G\left(T\right)⊊X\end{array}}{\min }\frac{\left|N_G\right(T\left)\right|}{\left|T\right|}\}$ otherwise. Fan and Lin [9] investigated $b\left(G\right)$ from spectral perspectives, and provided tight sufficient conditions in terms of the spectral radius of a graph G to guarantee $b\left(G\right)⩾r$, where r is a positive integer. The study of the existence of k-factors in graphs is a classic problem in graph theory. Fan and Lin [9] also provided the spectral radius conditions for 1-binding graphs to contain a perfect matching and a 2-factor, respectively. In this paper, we consider the bipartite analogues of those results obtained in [9]. For a balanced bipartite graph G, we first provide two sufficient conditions to guarantee that $b^{\prime }\left(G\right)⩾r$ for a positive integer r, in which one is based on the size, the other is based on the spectral radius of G. Then we establish two sufficient conditions for the balanced bipartite graph G with $\delta \left(G\right)⩾k$ to admit a k-factor via the size and spectral radius of G, respectively. Related problems are also proposed for potential future work.