Comparing list-color functions of uniform hypergraphs with their chromatic polynomials
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866917863230537728 |
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| author | Dong, Fengming Zhang, Meiqiao |
| author_facet | Dong, Fengming Zhang, Meiqiao |
| contents | In [J. Combin. Theory Ser. B 161 (2023), 109--119], the authors showed that the list-color function $P_l(G,k)$ of any simple graph $G$ of size $m$ coincides with its chromatic polynomial $P(G,k)$ for all integers $k\ge m-1$. In this article, we extend this conclusion to any uniform hypergraph. Furthermore, we show that for any $r$-uniform hypergraph ${\cal H}=(V,E)$, where $r\ge 2$, $P({\cal H}, L)-P({\cal H},k)\ge (k-|E|+1)k^{|V|-r-1}\sum\limits_{e\in E}\left (k-\left|\bigcap\limits_{v\in e}L(v)\right|\right )$ holds for all integers $k$ with $k\ge |E|-1\ge 4$ and all $k$-assignments $L$ of ${\cal H}$, where $P({\cal H}, L)$ is the number of $L$-colorings of ${\cal H}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_02497 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Comparing list-color functions of uniform hypergraphs with their chromatic polynomials Dong, Fengming Zhang, Meiqiao Combinatorics 05C15, 05C30, 05C31 In [J. Combin. Theory Ser. B 161 (2023), 109--119], the authors showed that the list-color function $P_l(G,k)$ of any simple graph $G$ of size $m$ coincides with its chromatic polynomial $P(G,k)$ for all integers $k\ge m-1$. In this article, we extend this conclusion to any uniform hypergraph. Furthermore, we show that for any $r$-uniform hypergraph ${\cal H}=(V,E)$, where $r\ge 2$, $P({\cal H}, L)-P({\cal H},k)\ge (k-|E|+1)k^{|V|-r-1}\sum\limits_{e\in E}\left (k-\left|\bigcap\limits_{v\in e}L(v)\right|\right )$ holds for all integers $k$ with $k\ge |E|-1\ge 4$ and all $k$-assignments $L$ of ${\cal H}$, where $P({\cal H}, L)$ is the number of $L$-colorings of ${\cal H}$. |
| title | Comparing list-color functions of uniform hypergraphs with their chromatic polynomials |
| topic | Combinatorics 05C15, 05C30, 05C31 |
| url | https://arxiv.org/abs/2305.02497 |