New Bounds for Chromatic Polynomials and Chromatic Roots

TL;DR

This paper presents improved bounds on chromatic polynomials and roots of graphs, refining existing inequalities and extending their applicability to dense graphs and specific graph classes.

Contribution

It introduces new upper bounds for chromatic polynomials based on maximum degree and connectivity, advancing the understanding of chromatic roots.

Findings

Improved bound:  \

Bound on chromatic roots for dense graphs is tighter than previous results.

Enhanced inequalities for connected k-chromatic graphs with k  4.

Abstract

If $G$ is a $k$-chromatic graph of order $n$ then it is known that the chromatic polynomial of $G$, $\pi(G,x)$, is at most $x(x-1)\cdots (x-(k-1))x^{n-k} = (x)_{\downarrow k}x^{n-k}$ for every $x\in \mathbb{N}$. We improve here this bound by showing that \[ \pi(G,x) \leq (x)_{\downarrow k} (x-1)^{\Delta(G)-k+1} x^{n-1-\Delta(G)}\] for every $x\in \mathbb{N},$ where $\Delta(G)$ is the maximum degree of $G$. Secondly, we show that if $G$ is a connected $k$-chromatic graph of order $n$ where $k\geq 4$ then $\pi(G,x)$ is at most $(x)_{\downarrow k}(x-1)^{n-k}$ for every real $x\geq n-2+\left( {n \choose 2} -{k \choose 2}-n+k \right)^2$ (it had been previously conjectured that this inequality holds for all $x \geq k$). Finally, we provide an upper bound on the moduli of the chromatic roots that is an improvment over known bounds for dense graphs.