Colorings v.s. list colorings of uniform hypergraphs

TL;DR

This paper extends known results on list colorings of graphs to hypergraphs, showing that for sufficiently large list sizes, the minimal number of colorings occurs under constant list assignments, using a refined broken cycle theorem.

Contribution

It introduces a refined broken cycle theorem for hypergraphs and establishes a threshold for list assignments where constant lists minimize colorings, extending prior graph results.

Findings

For large k, constant list assignments minimize colorings.

Established a threshold k > 1.135(m-1) for hypergraphs.

Extended graph coloring results to hypergraphs.

Abstract

Let $r$ be an integer with $r\ge 2$ and $G$ be a connected $r$-uniform hypergraph with $m$ edges. By refining the broken cycle theorem for hypergraphs, we show that if $k>\frac{m-1}{\ln(1+\sqrt{2})}\approx 1.135 (m-1)$ then the $k$-list assignment of $G$ admitting the fewest colorings is the constant list assignment. This extends the previous results of Donner, Thomassen and the current authors for graphs.