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

TL;DR

The paper proves that every cyclically $3$-edge-connected cubic graph not containing a Klee-graph admits two perfect matchings whose union leaves an acyclic subgraph, addressing a strengthened form of Mazzuoccolo's conjectures toward Fan–Raspaud and Berge–Fulkerson. The authors show that for any edge $e$ and any $1^+$-factor $F$, there is a perfect matching $M$ with $e\in M$ such that $G\setminus (F\cup M)$ is acyclic, via an equivalent reformulation involving collections of disjoint circuits. The core of the work is a contradiction-based induction that eliminates possible minimal counterexamples by a suite of reductions and distance-2 arguments, together with a careful exclusion of Klee-graphs. The results extend understanding of how higher connectivity constrains perfect-matchings and circuits in cubic graphs, contributing steps toward resolving long-standing conjectures such as the Fan–Raspaud and Berge–Fulkerson conjectures. The findings also clarify the role of Klee-graphs as exceptional cases and establish a framework for further generalizations.

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.

Figures (12)

• Figure 1: Examples of Klee-graphs on 4 upto 12 vertices, left to right.
• Figure 2: An example of a Klee ladder, $KL_{12}$. There is a unique perfect matching (here depicted using dotted lines) containing the edge $e$. The complement of this perfect matching is a Hamiltonian circuit.
• Figure 3: An illustration of a Klee ladder, a ladder, a Möbius ladder, and a quasi-ladder.
• Figure 4: An example of a Hamiltonian circuit $C_e$ (drawn using double lines) avoiding a given edge $e$ whose complement is a perfect matching $M_e$ containing $e$, for both possible positions of the prescribed edge $e$ in a ladder (top line) or a Möbius ladder (bottom line). A second perfect matching $M'_e$ can be obtained by the symmetric difference with the grey 4-circuit.
• Figure 5: An example of a Hamiltonian circuit $C_e$ avoiding a given edge $e$ (drawn using double lines) whose complement is a perfect matching $M_e$ containing $e$, for an edge $e$ contained in the grid $P_2\square P_{k-2}$ (top line) and in the 4-circuit outside the grid (bottom line) of a quasi-ladder. A second perfect matching $M'_e$ can be obtained by the symmetric difference with the grey 4-circuit.

Theorems & Definitions (18)

• Conjecture 1.1: Mazzuoccolo, 2013 MazzuoccoloS4
• Theorem 1.2: Kardoš, Máčajová & Zerafa, 2023 quelling1
• Conjecture 1.3: Mazzuoccolo, 2013 MazzuoccoloS4
• Conjecture 1.4: Mazzuoccolo, 2013 MazzuoccoloS4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1: klee dmtcs
• Lemma 2.2: klee dmtcs
• Proposition 2.3
• Proposition 2.4