Let $G$ be a bipartite graph with bipartition $(X,Y)$. Inspired by a hypergraph problem posed by Kostochka et al. (2021), 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.
@article{10_37236_12462,
author = {Mikhail Lavrov and Jennifer Vandenbussche},
title = {A {Hall-type} condition for path covers in bipartite graphs},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {3},
doi = {10.37236/12462},
zbl = {1548.05274},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12462/}
}
TY - JOUR
AU - Mikhail Lavrov
AU - Jennifer Vandenbussche
TI - A Hall-type condition for path covers in bipartite graphs
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/12462/
DO - 10.37236/12462
ID - 10_37236_12462
ER -
%0 Journal Article
%A Mikhail Lavrov
%A Jennifer Vandenbussche
%T A Hall-type condition for path covers in bipartite graphs
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/12462/
%R 10.37236/12462
%F 10_37236_12462
Mikhail Lavrov; Jennifer Vandenbussche. A Hall-type condition for path covers in bipartite graphs. The electronic journal of combinatorics, Tome 31 (2024) no. 3. doi: 10.37236/12462
