The Frank number and nowhere-zero flows on graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910754432614400 |
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| author | Goedgebeur, Jan Máčajová, Edita Renders, Jarne |
| author_facet | Goedgebeur, Jan Máčajová, Edita Renders, Jarne |
| contents | An edge $e$ of a graph $G$ is called deletable for some orientation $o$ if the restriction of $o$ to $G-e$ is a strong orientation. Inspired by a problem of Frank, in 2021 Hörsch and Szigeti proposed a new parameter for $3$-edge-connected graphs, called the Frank number, which refines $k$-edge-connectivity. The Frank number is defined as the minimum number of orientations of $G$ for which every edge of $G$ is deletable in at least one of them. They showed that every $3$-edge-connected graph has Frank number at most $7$ and that in case these graphs are also $3$-edge-colourable the parameter is at most $3$. Here we strengthen both results by showing that every $3$-edge-connected graph has Frank number at most $4$ and that every graph which is $3$-edge-connected and $3$-edge-colourable has Frank number $2$. The latter also confirms a conjecture by Barát and Blázsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number $2$ and use them in an algorithm to computationally show that the Petersen graph is the only cyclically $4$-edge-connected cubic graph up to $36$ vertices having Frank number greater than $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_02133 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Frank number and nowhere-zero flows on graphs Goedgebeur, Jan Máčajová, Edita Renders, Jarne Combinatorics Discrete Mathematics An edge $e$ of a graph $G$ is called deletable for some orientation $o$ if the restriction of $o$ to $G-e$ is a strong orientation. Inspired by a problem of Frank, in 2021 Hörsch and Szigeti proposed a new parameter for $3$-edge-connected graphs, called the Frank number, which refines $k$-edge-connectivity. The Frank number is defined as the minimum number of orientations of $G$ for which every edge of $G$ is deletable in at least one of them. They showed that every $3$-edge-connected graph has Frank number at most $7$ and that in case these graphs are also $3$-edge-colourable the parameter is at most $3$. Here we strengthen both results by showing that every $3$-edge-connected graph has Frank number at most $4$ and that every graph which is $3$-edge-connected and $3$-edge-colourable has Frank number $2$. The latter also confirms a conjecture by Barát and Blázsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number $2$ and use them in an algorithm to computationally show that the Petersen graph is the only cyclically $4$-edge-connected cubic graph up to $36$ vertices having Frank number greater than $2$. |
| title | The Frank number and nowhere-zero flows on graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2305.02133 |