Three-cuts are a charm: acyclicity in 3-connected cubic graphs

TL;DR

This paper proves that every cyclically 3-edge-connected cubic graph satisfies a conjecture related to perfect matchings and acyclicity, advancing understanding of longstanding graph theory conjectures.

Contribution

The authors prove that cyclically 3-edge-connected cubic graphs satisfy a conjecture about perfect matchings and acyclicity, extending previous results and partially resolving related conjectures.

Findings

Proved the $S_4$-Conjecture as a theorem.

Established that such graphs satisfy the acyclic subgraph conjecture.

Connected the results to broader conjectures in graph theory.

Abstract

Let $G$ be a bridgeless cubic graph. In 2023, the three authors solved a conjecture (also known as the $S_4$-Conjecture) made by Mazzuoccolo in 2013: there exist two perfect matchings of $G$ such that the complement of their union is a bipartite subgraph of $G$. They actually show that given any $1^+$-factor $F$ (a spanning subgraph of $G$ such that its vertices have degree at least 1) and an arbitrary edge $e$ of $G$, there exists a perfect matching $M$ of $G$ containing $e$ such that $G\setminus (F\cup M)$ is bipartite. This is a step closer to comprehend better the Fan--Raspaud Conjecture and eventually the Berge--Fulkerson Conjecture. The $S_4$-Conjecture, now a theorem, is also the weakest assertion in a series of three conjectures made by Mazzuoccolo in 2013, with the next stronger statement being: there exist two perfect matchings of $G$ such that the complement of their union is an acyclic subgraph of $G$. Unfortunately, this conjecture is not true: Jin, Steffen, and Mazzuoccolo later showed that there exists a counterexample admitting 2-cuts. Here we show that, despite of this, every cyclically 3-edge-connected cubic graph satisfies this second conjecture.