On the degree-chromatic polynomial of a tree

TL;DR

This paper proves a conjecture regarding the leading terms of the degree chromatic polynomial for trees, which counts colorings with restrictions on adjacent same-colored vertices.

Contribution

It confirms Humpert and Martin's conjecture on the asymptotic behavior of the degree chromatic polynomial for trees.

Findings

Proof of Humpert and Martin's conjecture

Characterization of the leading terms of the polynomial

Enhanced understanding of constrained graph colorings

Abstract

The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree chromatic polynomial of a tree.