A Hall-type condition for path covers in bipartite graphs

TL;DR

This paper proposes a Hall-type condition for path covers in bipartite graphs, providing new bounds and verifying the conjecture in special graph classes such as forests, degree-3 graphs, and high-girth regular graphs.

Contribution

It introduces a novel Hall-type condition for path covers in bipartite graphs and proves its validity in specific graph classes, advancing understanding of path cover problems.

Findings

Conjecture holds for forests

Conjecture holds for graphs with maximum degree 3

Conjecture holds for regular graphs with high girth

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.