Strong binding numbers and factors

Let $G$ be a simple graph. The $k$-th neighborhood of a vertex subset $S\subseteq V(G)$, denoted ${\mathrm{\Lambda }}^k(S)$, is the set of vertices that are adjacent to at least $k$ vertices in $S$. The $k$-th binding number ${\beta }^k(G)$ is defined as the minimum ratio $\mathrm{\mid }{\mathrm{\Lambda }}^k(S)\mathrm{\mid }\mathrm{/}\mathrm{\mid }S\mathrm{\mid }$ over all subsets $S\subseteq V(G)$ with $\mathrm{\mid }S\mathrm{\mid }\ge k$ and ${\mathrm{\Lambda }}^k(S)\mathrm{\ne }V(G)$. This parameter generalizes the classical binding number introduced by Woodall. Andersen showed that the condition ${\beta }^1(G)\ge 1$ does not guarantee the existence of a $1$-factor in $G$, while Barát et al. proved that ${\beta }^2(G)\ge 1$ 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 $k\mathrm{\mid }V(G)\mathrm{\mid }$ and ${\beta }^k(G)\ge 1$ 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 ${\beta }^k(G)\ge 1$ 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 ${\beta }^k(G,X)\ge 1$, the graph admits $k$ disjoint matchings, each covering $X$.