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arXiv · 2310.02909

The double Hall property and cycle covers in bipartite graphs

János Barát·Andrzej Grzesik·Attila Jung·Zoltán Lóránt Nagy·Dömötör Pálvölgyi
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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.

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János Barát, Andrzej Grzesik, Attila Jung, Zoltán Lóránt Nagy, Dömötör Pálvölgyi. 2023-10-04. The double Hall property and cycle covers in bipartite graphs. https://arxiv.org/abs/2310.02909

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