Computational methods for finding bi-regular cages
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913586129928192 |
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| author | Goedgebeur, Jan Jooken, Jorik Eede, Tibo Van den |
| author_facet | Goedgebeur, Jan Jooken, Jorik Eede, Tibo Van den |
| contents | An $(\{r,m\};g)$-graph is a (simple, undirected) graph of girth $g\geq3$ with vertices of degrees $r$ and $m$ where $2 \leq r < m$ . Given $r,m,g$, we seek the $(\{r,m\};g)$-graphs of minimum order, called $(\{r,m\};g)$-cages or bi-regular cages, whose order is denoted by $n(\{r,m\};g)$. In this paper, we use computational methods for finding $(\{r,m\};g)$-graphs of small order. Firstly, we present an exhaustive generation algorithm, which leads to $\unicode{x2013}$ previously unknown $\unicode{x2013}$ exhaustive lists of $(\{r,m\};g)$-cages for 24 different triples $(r,m,g)$. This also leads to the improvement of the lower bound of $n(\{4,5\};7)$ from 66 to 69. Secondly, we improve 49 upper bounds of $n(\{r,m\};g)$ based on constructions that start from $r$-regular graphs. Lastly, we generalize a theorem by Aguilar, Araujo-Pardo and Berman [arXiv:2305.03290, 2023], leading to 73 additional improved upper bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computational methods for finding bi-regular cages Goedgebeur, Jan Jooken, Jorik Eede, Tibo Van den Combinatorics Discrete Mathematics 05C07, 05C35, 05C85, 68R10, 90C35 An $(\{r,m\};g)$-graph is a (simple, undirected) graph of girth $g\geq3$ with vertices of degrees $r$ and $m$ where $2 \leq r < m$ . Given $r,m,g$, we seek the $(\{r,m\};g)$-graphs of minimum order, called $(\{r,m\};g)$-cages or bi-regular cages, whose order is denoted by $n(\{r,m\};g)$. In this paper, we use computational methods for finding $(\{r,m\};g)$-graphs of small order. Firstly, we present an exhaustive generation algorithm, which leads to $\unicode{x2013}$ previously unknown $\unicode{x2013}$ exhaustive lists of $(\{r,m\};g)$-cages for 24 different triples $(r,m,g)$. This also leads to the improvement of the lower bound of $n(\{4,5\};7)$ from 66 to 69. Secondly, we improve 49 upper bounds of $n(\{r,m\};g)$ based on constructions that start from $r$-regular graphs. Lastly, we generalize a theorem by Aguilar, Araujo-Pardo and Berman [arXiv:2305.03290, 2023], leading to 73 additional improved upper bounds. |
| title | Computational methods for finding bi-regular cages |
| topic | Combinatorics Discrete Mathematics 05C07, 05C35, 05C85, 68R10, 90C35 |
| url | https://arxiv.org/abs/2411.17351 |