Strong binding numbers and factors

TL;DR

This paper generalizes the concept of binding numbers in graphs to establish conditions for the existence of k-factors, disjoint matchings, and extensions in bipartite graphs, broadening understanding of graph factorization.

Contribution

It extends previous results by proving that graphs with certain binding number conditions contain k-factors and multiple disjoint matchings, including new bounds for split and bipartite graphs.

Findings

Graphs with $eta^k(G) \,\ge\, 1$ contain k-factors if even order conditions are met.

Split graphs with even order and $eta^k(G) \,\ge\, 1$ admit (k+1)-factors.

Bipartite graphs with $eta^k(G, X) \,\ge\, 1$ have k disjoint matchings covering X.

Abstract

Let $G$ be a simple graph. The $k$-th neighborhood of a vertex subset $S \subseteq V(G)$, denoted $\Lambda^k(S)$, 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 $|\Lambda^k(S)|/|S|$ over all subsets $S \subseteq V(G)$ with $|S| \ge k$ and $\Lambda^k(S) \ne V(G)$. 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 Bar\'at 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 $k|V(G)|$ 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 $X$.