On rigid regular graphs and a problem of Babai and Pultr
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915154350833664 |
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| author | Knauer, Kolja Surroca, Gil Puig i |
| author_facet | Knauer, Kolja Surroca, Gil Puig i |
| contents | A graph is \textit{rigid} if it only admits the identity endomorphism. We show that for every $d\ge 3$ there exist infinitely many mutually rigid $d$-regular graphs of arbitrary odd girth $g\geq 7$. Moreover, we determine the minimum order of a rigid $d$-regular graph for every $d\ge 3$. This provides strong positive answers to a question of van der Zypen [https://mathoverflow.net/q/296483, https://mathoverflow.net/q/321108]. Further, we use our construction to show that every finite monoid is isomorphic to the endomorphism monoid of a regular graph. This solves a problem of Babai and Pultr [J. Comb.~Theory, Ser.~B, 1980]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11421 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On rigid regular graphs and a problem of Babai and Pultr Knauer, Kolja Surroca, Gil Puig i Combinatorics Discrete Mathematics A graph is \textit{rigid} if it only admits the identity endomorphism. We show that for every $d\ge 3$ there exist infinitely many mutually rigid $d$-regular graphs of arbitrary odd girth $g\geq 7$. Moreover, we determine the minimum order of a rigid $d$-regular graph for every $d\ge 3$. This provides strong positive answers to a question of van der Zypen [https://mathoverflow.net/q/296483, https://mathoverflow.net/q/321108]. Further, we use our construction to show that every finite monoid is isomorphic to the endomorphism monoid of a regular graph. This solves a problem of Babai and Pultr [J. Comb.~Theory, Ser.~B, 1980]. |
| title | On rigid regular graphs and a problem of Babai and Pultr |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2502.11421 |