Spectral conditions for $k$-extendability and $k$-factors of bipartite graphs

TL;DR

This paper establishes spectral conditions that guarantee the existence of $k$-extendability and $k$-factors in balanced bipartite graphs, generalizing previous results on perfect matchings and regular factors.

Contribution

It provides new spectral criteria for $k$-extendability and $k$-factors in bipartite graphs, extending prior work to more general cases.

Findings

Spectral conditions for $k$-extendability in bipartite graphs.

Spectral conditions for the existence of $k$-factors.

Extension of previous results on perfect matchings and regular factors.

Abstract

Let $G$ be a connected graph. If $G$ contains a matching of size $k$, and every matching of size $k$ is contained in a perfect matching of $G$, then $G$ is said to be \emph{$k$-extendable}. A $k$-regular spanning subgraph of $G$ is called a \textit{$k$-factor}. In this paper, we provide spectral conditions for a (balanced bipartite) graph with minimum degree $\delta$ to be $k$-extendable, and for the existence of a $k$-factor in a balanced bipartite graph, respectively. Our results generalize some previous results on perfect matchings of graphs, and extend the results in \cite{D.F} and \cite{W.L} to $k$-extendable graphs. Furthermore, our results generalize the result of Lu, Liu and Tian \cite{Lu-Liu} to general regular factors. Additionally, using the equivalence of $k$ edge-disjoint perfect matchings and $k$-factors in balanced bipartite graphs, our results can derive a spectral condition for the existence of $k$ edge-disjoint perfect matchings in balanced bipartite graphs.