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arXiv · 2310.05248

A Hall-type condition for path covers in bipartite graphs

Mikhail Lavrov·Jennifer Vandenbussche
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Abstract

Let $G$ be a bipartite graph with bipartition $(X,Y)$. Inspired by a hypergraph problem, we seek an upper bound on the number of disjoint paths needed to cover all the vertices of $X$. We conjecture that a Hall-type sufficient condition holds based on the maximum value of $|S|-|\mathsf\Lambda(S)|$, where $S\subseteq X$ and $\mathsf\Lambda(S)$ is the set of all vertices in $Y$ with at least two neighbors in $S$. This condition is also a necessary one for a hereditary version of the problem, where we delete vertices from $X$ and try to cover the remaining vertices by disjoint paths. The conjecture holds when $G$ is a forest, has maximum degree $3$, or is regular with high girth, and we prove those results in this paper.

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Mikhail Lavrov, Jennifer Vandenbussche. 2023-10-08. A Hall-type condition for path covers in bipartite graphs. https://arxiv.org/abs/2310.05248

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