An improved lower bound of $P(G, L)-P(G,k)$ for $k$-assignments $L$

TL;DR

This paper establishes a new lower bound for the difference between the number of $L$-colorings and $k$-colorings of a graph, improving understanding of graph coloring polynomials for graphs with certain properties.

Contribution

It introduces a novel lower bound for $P(G,L)-P(G,k)$ when $k geq m-1$, extending previous bounds and considering specific graph classes like $K_3$-free graphs.

Findings

For $k \\ge m-1 \\ge 3$, the difference $P(G,L)-P(G,k)$ has a new explicit lower bound.

The bound depends on the structure of the graph and the assignment $L$, with specific constants for $K_3$-free graphs.

When $k \\ge m-1$, it guarantees $P(G,L) \\ge P(G,k)$, ensuring $L$-colorings are at least as numerous as $k$-colorings.

Abstract

Let $G=(V,E)$ be a simple graph with $n$ vertices and $m$ edges, $P(G,k)$ be the chromatic polynomial of $G$, and $P(G,L)$ be the number of $L$-colorings of $G$ for any $k$-assignment $L$. In this article, we show that when $k\ge m-1\ge 3$, $P(G,L)-P(G,k)$ is bounded below by $\left ( (k-m+1)k^{n-3}+\frac{(k-m+3)c}{3} k^{n-5}\right )\sum\limits_{uv\in E}|L(u)\setminus L(v)|$, where $c\ge \frac{(m-1)(m-3)}{8}$, and in particular, if $G$ is $K_3$-free, then $c\ge {m-2\choose 2}+2\sqrt m-3$. Consequently, $P(G,L)\ge P(G,k)$ whenever $k\ge m-1$.