Tree Matchings

TL;DR

This paper establishes precise conditions under which bipartite graphs contain specific structured matchings called $(s,t)$-matchings, generalizing and confirming a conjecture in the field.

Contribution

The paper provides sharp criteria for the existence of $(s,t)$-matchings in bipartite graphs, including a proof of a previously conjectured case.

Findings

Derived sharp conditions for $(s,t)$-matchings

Confirmed a conjecture by Bonacina et al.

Extended understanding of structured matchings in bipartite graphs

Abstract

An $(s,t)$-matching in a bipartite graph $G=(U,V,E)$ is a subset of the edges $F$ such that each component of $G[F]$ is a tree with at most $t$ edges and each vertex in $U$ has $s$ neighbours in $G[H]$. We give sharp conditions for a bipartite graph to contain an $(s,t)$-matching. As a special case, we prove a conjecture of Bonacina, Galesi, Huynh and Wollan \cite{CNF}.