A Simple Condition for the Existence of Transversals

TL;DR

This paper introduces a new sufficient condition based on set sizes and intersections for the existence of transversals, and applies it to bipartite graphs without 4-cycles to guarantee matchings.

Contribution

It provides a novel sufficient condition for transversals in set families and applies it to bipartite graphs to ensure large matchings under certain degree constraints.

Findings

A new sufficient condition for the existence of transversals.

Application of the condition to bipartite graphs without 4-cycles.

Guarantee of a perfect matching under degree conditions.

Abstract

Hall's Theorem is a basic result in Combinatorics which states that the obvious necesssary condition for a finite family of sets to have a transversal is also sufficient. We present a sufficient (but not necessary) condition on the sizes of the sets in the family and the sizes of their intersections so that a transversal exists. Using this, we prove that in a bipartite graph $G$ (bipartition $\{A, B\}$), without 4-cycles, if $\deg(v) \geq \sqrt{2e|A|}$ for all $v \in A$, then $G$ has a matching of size $|A|$.