A rooted variant of Stanley's chromatic symmetric function

TL;DR

This paper introduces a rooted variant of Stanley's chromatic symmetric function, proves its irreducibility and distinguishes rooted trees, and provides combinatorial interpretations of related polynomials.

Contribution

It extends Stanley's chromatic symmetric function to rooted graphs, proves irreducibility for large N, confirms a rooted version of Stanley's conjecture, and interprets pointed chromatic functions combinatorially.

Findings

Rooted chromatic polynomials are irreducible for large N.

Rooted trees are uniquely identified by their rooted chromatic polynomials.

A specialized one-variable polynomial distinguishes rooted trees.

Abstract

Richard Stanley defined the chromatic symmetric function $X_G$ of a graph $G$ and asked whether there are non-isomorphic trees $T$ and $U$ with $X_T=X_U$. We study variants of the chromatic symmetric function for rooted graphs, where we require the root vertex to either use or avoid a specified color. We present combinatorial identities and recursions satisfied by these rooted chromatic polynomials, explain their relation to pointed chromatic functions and rooted $U$-polynomials, and prove three main theorems. First, for all non-empty connected graphs $G$, Stanley's polynomial $X_G(x_1,\ldots,x_N)$ is irreducible in $\mathbb{Q}[x_1,\ldots,x_N]$ for all large enough $N$. The same result holds for our rooted variant where the root node must avoid a specified color. We prove irreducibility by a novel combinatorial application of Eisenstein's Criterion. Second, we prove the rooted version of Stanley's Conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal. In fact, we prove that a one-variable specialization of the rooted chromatic polynomial (obtained by setting $x_0=x_1=q$, $x_2=x_3=1$, and $x_n=0$ for $n>3$) already distinguishes rooted trees. Third, we answer a question of Pawlowski by providing a combinatorial interpretation of the monomial expansion of pointed chromatic functions.