Three matching intersection property for matching covered graphs

TL;DR

This paper investigates the three matching intersection property in matching covered graphs, providing a necessary and sufficient condition and applying it to characterize specific graph classes like Halin and 4-regular graphs.

Contribution

It introduces a precise condition for the three matching intersection property and applies it to characterize certain classes of matching covered graphs.

Findings

Established a necessary and sufficient condition for the 3PM property.

Characterized the 3PM property in Halin graphs.

Analyzed the 3PM property in 4-regular graphs.

Abstract

In connection with Fulkerson's conjecture on cycle covers, Fan and Raspaud proposed a weaker conjecture: For every bridgeless cubic graph $G$, there are three perfect matchings $M_1$, $M_2$, and $M_3$ such that $M_1\cap M_2 \cap M_3=\emptyset$. We call the property specified in this conjecture the three matching intersection property (and 3PM property for short). We study this property on matching covered graphs. The main results are a necessary and sufficient condition and its applications to characterization of special graphs, such as the Halin graphs and 4-regular graphs.