Chromatic Polynomial Evaluation Spectra

TL;DR

This paper demonstrates that the set of chromatic polynomial values for graphs on n vertices grows exponentially, confirming a conjecture for q=3 and extending previous results to all fixed real q not equal to 0, 1, or 2.

Contribution

It proves that the number of distinct chromatic polynomial values for graphs on n vertices is exponential in n for all fixed real q ≠ 0,1,2, confirming a conjecture for q=3.

Findings

Number of chromatic polynomial values grows exponentially with n.

Confirmed the conjecture for q=3.

Extended results to all fixed real q ≠ 0,1,2.

Abstract

Around 10 years ago, Agol and Krushkal showed that the number of chromatic polynomials $P_{G}$ arising from graphs $G$ on $n$ vertices grows exponentially with $n$, by establishing that the (dual) flow polynomial $F_{G}\left(\frac{3+\sqrt{5}}{2}\right)$ already takes on exponentially many values, if one varies $G$ over all planar cubic graphs $G$ on $n$ vertices. We show, more generally, that the size of the set $\{P_G(q): |V(G)|=n\}$ is exponential in $n$, for every fixed real number $q \neq 0,1,2$. In fact, our approach can also be pushed to show that $P_{G}(q)$ already takes on exponentially many values, if we only vary $G$ over all planar graphs on $n$ vertices. The case $q=3$ confirms a conjecture of Agol, which was initially motivated by the $\mathsf{NP}$-completeness of planar $3$-colorability.