A class of trees determined by their chromatic symmetric functions

Abstract

Stanley introduced the concept of chromatic symmetric functions of graphs which extends and refines the notion of chromatic polynomials of graphs, and asked whether trees are determined up to isomorphism by their chromatic symmetric functions. Using the technique of differentiation with respect to power-sum symmetric functions, we give a positive answer to Stanley's question for the class of trees with exactly two vertices of degree at least 3. In addition, we prove that for any tree $T$, the generalized degree sequence for subtrees of $T$ is determined by the chromatic symmetric function of $T$, providing evidence to a conjecture of Crew.

Figures (1)

• Figure 1: Two unicyclic graphs with the same chromatic symmetric function

Theorems & Definitions (18)

• Conjecture 1.1: Stanley's tree problem, 1995
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Lemma 2.4
• Proposition 2.5
• Corollary 2.6
• Lemma 3.1
• proof