On the Girth of Graph Lifts
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917167709028352 |
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| author | Hoory, Shlomo |
| author_facet | Hoory, Shlomo |
| contents | The size of the smallest $k$-regular graph of girth at least $g$ is denoted by the well-studied function $n(k,g)$. We introduce an analogous function $n(H,g)$, defined as the smallest size graph of girth at least $g$ that is a lift (or cover) of the, possibly non-regular, graph $H$. We prove that the two main combinatorial bounds on $n(k,g)$ -- the Moore lower bound and the Erdös-Sachs upper bound -- carry over to the new lift setting.
We also consider two other functions: i) The smallest size graph of girth at least $g$ sharing a universal cover with $H$. We prove that it is the same as $n(H,g)$ up to a multiplicative constant. ii) The smallest size graph of girth least $g$ with a prescribed degree distribution. We discuss this known generalization and argue that the new suggested definitions are superior.
We conclude with experimental results for a specific base graph, followed by conjectures and open problems for future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01238 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Girth of Graph Lifts Hoory, Shlomo Combinatorics The size of the smallest $k$-regular graph of girth at least $g$ is denoted by the well-studied function $n(k,g)$. We introduce an analogous function $n(H,g)$, defined as the smallest size graph of girth at least $g$ that is a lift (or cover) of the, possibly non-regular, graph $H$. We prove that the two main combinatorial bounds on $n(k,g)$ -- the Moore lower bound and the Erdös-Sachs upper bound -- carry over to the new lift setting. We also consider two other functions: i) The smallest size graph of girth at least $g$ sharing a universal cover with $H$. We prove that it is the same as $n(H,g)$ up to a multiplicative constant. ii) The smallest size graph of girth least $g$ with a prescribed degree distribution. We discuss this known generalization and argue that the new suggested definitions are superior. We conclude with experimental results for a specific base graph, followed by conjectures and open problems for future research. |
| title | On the Girth of Graph Lifts |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.01238 |