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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| Accesso online: | https://arxiv.org/abs/2209.15453 |
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| _version_ | 1866915191890903040 |
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| author | Knauer, Kolja Surroca, Gil Puig i |
| author_facet | Knauer, Kolja Surroca, Gil Puig i |
| contents | We show that every commutative idempotent monoid (a.k.a lattice) is the endomorphism monoid of a subcubic graph. This solves a problem of Babai and Pultr [J. Comb.~Theory, Ser.~B, 1980] and the degree bound is best-possible. On the other hand, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids. As a by-product we prove that monoids can be represented by graphs of bounded expansion (reproving a result of Nešetřil and Ossona de Mendez) and $k$-cancellative monoids can be represented by graphs of bounded degree. Finally, we show that not all completely regular monoids can be represented by graphs excluding topological minor (strengthening a result of Babai and Pultr). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_15453 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On endomorphism universality of sparse graph classes Knauer, Kolja Surroca, Gil Puig i Combinatorics We show that every commutative idempotent monoid (a.k.a lattice) is the endomorphism monoid of a subcubic graph. This solves a problem of Babai and Pultr [J. Comb.~Theory, Ser.~B, 1980] and the degree bound is best-possible. On the other hand, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids. As a by-product we prove that monoids can be represented by graphs of bounded expansion (reproving a result of Nešetřil and Ossona de Mendez) and $k$-cancellative monoids can be represented by graphs of bounded degree. Finally, we show that not all completely regular monoids can be represented by graphs excluding topological minor (strengthening a result of Babai and Pultr). |
| title | On endomorphism universality of sparse graph classes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2209.15453 |