Number of independent transversals in multipartite graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913778001510400 |
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| author | Tang, Yantao Zhao, Yi |
| author_facet | Tang, Yantao Zhao, Yi |
| contents | An independent transversal in a multipartite graph is an independent set that intersects each part in exactly one vertex. We show that for every even integer $r\ge 2$, there exist $c_r>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)-t$, where $t=o(n)$, contains at least $c_r t n^{r-1}$ independent transversals. This is best possible up to the value of $c_r$. Our result confirms a conjecture of Haxell and Szabó from 2006 and partially answers a question raised by Erdős in 1972 and studied by Bollobás, Erdős and Szemerédi in 1975. We also show that, given any integer $s\ge 2$ and even integer $r\ge 2$, there exist $c_{r,s}>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)- c_{r, s} n^{1-1/s}$ contains an independent set with exactly $s$ vertices in each part. This is best possible up to the value of $c_{r, s}$ if a widely believed conjecture for the Zarankiewicz number holds. Our result partially answers a question raised by Di Braccio and Illingworth recently. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03950 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Number of independent transversals in multipartite graphs Tang, Yantao Zhao, Yi Combinatorics 05C69, 05C35 (Primary) 05D15, 05C07, 05C30 (Secondary) An independent transversal in a multipartite graph is an independent set that intersects each part in exactly one vertex. We show that for every even integer $r\ge 2$, there exist $c_r>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)-t$, where $t=o(n)$, contains at least $c_r t n^{r-1}$ independent transversals. This is best possible up to the value of $c_r$. Our result confirms a conjecture of Haxell and Szabó from 2006 and partially answers a question raised by Erdős in 1972 and studied by Bollobás, Erdős and Szemerédi in 1975. We also show that, given any integer $s\ge 2$ and even integer $r\ge 2$, there exist $c_{r,s}>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)- c_{r, s} n^{1-1/s}$ contains an independent set with exactly $s$ vertices in each part. This is best possible up to the value of $c_{r, s}$ if a widely believed conjecture for the Zarankiewicz number holds. Our result partially answers a question raised by Di Braccio and Illingworth recently. |
| title | Number of independent transversals in multipartite graphs |
| topic | Combinatorics 05C69, 05C35 (Primary) 05D15, 05C07, 05C30 (Secondary) |
| url | https://arxiv.org/abs/2504.03950 |