On two consequences of Berge-Fulkerson conjecture

TL;DR

This paper explores two implications of the Berge-Fulkerson conjecture related to perfect matchings in bridgeless cubic graphs, establishing equivalences and properties of potential counterexamples.

Contribution

It demonstrates the equivalence of one statement to the Fan-Raspaud conjecture and characterizes the smallest counterexample to the other statement.

Findings

First statement is equivalent to Fan-Raspaud conjecture

Smallest counterexample to second statement is cyclically 4-edge-connected

Provides insights into structure of potential counterexamples

Abstract

The classical Berge-Fulkerson conjecture states that any bridgeless cubic graph $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to two of the perfect matchings from the list. In this short note, we discuss two statements that are consequences of this conjecture. We show that the first statement is equivalent to Fan-Raspaud conjecture. We also show that the smallest counter-example to the second one is a cyclically $4$-edge-connected cubic graph.