Exponentially Many 4-List-Colorings of Triangle-Free Graphs on Surfaces

TL;DR

This paper extends lower bounds on the number of list colorings in graphs embedded on surfaces, proving exponential growth for triangle-free graphs with 4-list-assignments, generalizing previous results for 3- and 5-list-assignments.

Contribution

It establishes a new exponential lower bound on the number of list colorings for triangle-free graphs on surfaces with 4-list-assignments, broadening the scope of prior work.

Findings

Proves exponential lower bounds for 4-list-colorings of triangle-free surface-embedded graphs.

Generalizes previous results from 3- and 5-list-colorings to 4-list-colorings.

Provides explicit constants for the lower bound: epsilon=1/8, alpha=130.

Abstract

Thomassen proved that every planar graph $G$ on $n$ vertices has at least $2^{n/9}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ and at least $2^{n/10000}$ distinct $L$-colorings if $L$ is a 3-list-assignment for $G$ and $G$ has girth at least five. Postle and Thomas proved that if $G$ is a graph on $n$ vertices embedded on a surface $\Sigma$ of genus $g$, then there exist constants $\epsilon,c_g > 0$ such that if $G$ has an $L$-coloring, then $G$ has at least $c_g2^{\epsilon n}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ or if $L$ is a 3-list-assignment for $G$ and $G$ has girth at least five. More generally, they proved that there exist constants $\epsilon,\alpha>0$ such that if $G$ is a graph on $n$ vertices embedded in a surface $\Sigma$ of fixed genus $g$, $H$ is a proper subgraph of $G$, and $\phi$ is an $L$-coloring of $H$ that extends to an $L$-coloring of $G$, then $\phi$ extends to at least $2^{\epsilon(n - \alpha(g + |V(H)|))}$ distinct $L$-colorings of $G$ if $L$ is a 5-list-assignment or if $L$ is a 3-list-assignment and $G$ has girth at least five. We prove the same result if $G$ is triangle-free and $L$ is a 4-list-assignment of $G$, where $\epsilon=\frac{1}{8}$, and $\alpha= 130$.