Three-cuts are a charm: acyclicity in 3-connected cubic graphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2023
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| _version_ | 1866915148537528320 |
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| author | Kardoš, František Máčajová, Edita Zerafa, Jean Paul |
| author_facet | Kardoš, František Máčajová, Edita Zerafa, Jean Paul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06944 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Three-cuts are a charm: acyclicity in 3-connected cubic graphs Kardoš, František Máčajová, Edita Zerafa, Jean Paul Combinatorics 05C15, 05C70 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. |
| title | Three-cuts are a charm: acyclicity in 3-connected cubic graphs |
| topic | Combinatorics 05C15, 05C70 |
| url | https://arxiv.org/abs/2309.06944 |