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

TL;DR

This paper establishes a lower bound on the difference between the chromatic polynomial and list-color function for uniform hypergraphs, leading to an improved upper bound on the parameter '() that measures their equality point.

Contribution

It provides a new lower bound for the difference between chromatic polynomials and list-color functions in uniform hypergraphs, improving the known bound on '().

Findings

Derived a lower bound for P(,L) - P(,k) for r-uniform hypergraphs.

Showed that '()  m-1, improving previous bounds.

Applicable for hypergraphs with r  3, order n, size m, and k  m-1.

Abstract

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 $\tau'({\cal H})$ 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 $\tau'({\cal H})\le m-1$, improving the current best result on $\tau'({\cal H})$.