The double Hall property and cycle covers in bipartite graphs

TL;DR

This paper investigates bipartite graphs with the double Hall property, proving the existence of a 2-factor covering one part and establishing bounds on edges, advancing understanding of cycle covers and degree restrictions.

Contribution

It proves the existence of a 2-factor covering part of bipartite graphs with the double Hall property and confirms Salia's conjecture under degree restrictions.

Findings

Existence of a 2-factor covering A in such graphs.

Validation of Salia's conjecture for graphs with degree restrictions.

A sharp lower bound on the number of edges in graphs with the double Hall property.

Abstract

In a graph $G$, the $2$-neighborhood of a vertex set $X$ consists of all vertices of $G$ having at least $2$ neighbors in $X$. We say that a bipartite graph $G(A,B)$ satisfies the double Hall property if $|A|\geq2$, and every subset $X \subseteq A$ of size at least $2$ has a $2$-neighborhood of size at least $|X|$. Salia conjectured that any bipartite graph $G(A,B)$ satisfying the double Hall property contains a cycle covering $A$. Here, we prove the existence of a $2$-factor covering $A$ in any bipartite graph $G(A,B)$ satisfying the double Hall property. We also show Salia's conjecture for graphs with restricted degrees of vertices in $B$. Additionally, we prove a lower bound on the number of edges in a graph satisfying the double Hall property, and the bound is sharp up to a constant factor.