Strong binding numbers and factors

Abstract

Let $G$ be a simple graph. The $k$-th neighborhood of a vertex subset $S\subseteq V(G)$, denoted $\hat{I}{›}^k(S)$, is the set of vertices that are adjacent to at least $k$ vertices in $S$. The $k$-th binding number $β^k(G)$ is defined as the minimum ratio $|\hat{I}{›}^k(S)|/|S|$ over all subsets $S\subseteq V(G)$ with $|S|\ge k$ and $\hat{I}{›}^k(S)\ne V(G)$. This parameter generalizes the classical binding number introduced by Woodall. Andersen showed that the condition $β^1(G) \ge 1$ does not guarantee the existence of a $1$-factor in $G$, while Barát et al. proved that $β^2(G) \ge 1$ suffices for the existence of a $2$-factor. In this paper, we extend this result to general $k\ge 2$ by showing that any graph $G$ with even $k|V(G)|$ and $β^k(G) \ge 1$ contains a $k$-factor. Moreover, if $G$ is additionally a split graph of even order, then it admits a $(k+1)$-factor. We also prove that any graph $G$ with $β^k(G) \ge 1$ contains at least $k-1$ disjoint perfect or near-perfect matchings. Finally, for any bipartite graph $G$ with bipartition $(X,Y)$, we introduce an analogue of the $k$-th binding number and show that, under the condition $β^k(G, X) \ge 1$, the graph admits $k$ disjoint matchings, each covering $X$.

21 pages, 1 figure