Comparing list-color functions of uniform hypergraphs with their chromatic polynomials (III)
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| Format: | Preprint |
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2022
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| _version_ | 1866916416082411520 |
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| author | Dong, Fengming Zhang, Meiqiao |
| author_facet | Dong, Fengming Zhang, Meiqiao |
| contents | For a hypergraph ${\cal H}$, let $P({\cal H},k)$ and $P_l({\cal H},k)$ be its chromatic polynomial and list-color function respectively, and let $τ'({\cal H})$ be the least non-negative integer $q$ such that $P({\cal H},k)=P_l({\cal H},k)$ holds for all integers $k\ge q$. In this article, we show that for any $r$-uniform hypergraph ${\cal H}$ of order $n$ and size $m$ and any $k$-assignment $L$ of ${\cal H}$, where $r\ge 3$, $P({\cal H},L)-P({\cal H},k)\ge \min \{0.02k, k-(m-1)\} k^{n-r-1}\sum_{e\in E({\cal H})} \left ( k-\left |\bigcap_{v\in e}L(v)\right | \right )$ holds for $k\ge m-1\ge 4$. It follows that $τ'({\cal H})\le m-1$, improving the current best result on $τ'({\cal H})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_02045 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Comparing list-color functions of uniform hypergraphs with their chromatic polynomials (III) Dong, Fengming Zhang, Meiqiao Combinatorics 05C15, 05C30, 05C31 For a hypergraph ${\cal H}$, let $P({\cal H},k)$ and $P_l({\cal H},k)$ be its chromatic polynomial and list-color function respectively, and let $τ'({\cal H})$ be the least non-negative integer $q$ such that $P({\cal H},k)=P_l({\cal H},k)$ holds for all integers $k\ge q$. In this article, we show that for any $r$-uniform hypergraph ${\cal H}$ of order $n$ and size $m$ and any $k$-assignment $L$ of ${\cal H}$, where $r\ge 3$, $P({\cal H},L)-P({\cal H},k)\ge \min \{0.02k, k-(m-1)\} k^{n-r-1}\sum_{e\in E({\cal H})} \left ( k-\left |\bigcap_{v\in e}L(v)\right | \right )$ holds for $k\ge m-1\ge 4$. It follows that $τ'({\cal H})\le m-1$, improving the current best result on $τ'({\cal H})$. |
| title | Comparing list-color functions of uniform hypergraphs with their chromatic polynomials (III) |
| topic | Combinatorics 05C15, 05C30, 05C31 |
| url | https://arxiv.org/abs/2212.02045 |