Comparing list-color functions of uniform hypergraphs with their chromatic polynomials

Abstract

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 ${\cal H}$.