A Note on Distinguishing Trees with the Chromatic Symmetric Function

TL;DR

This paper demonstrates that the chromatic symmetric function of a tree uniquely determines the size of its core subtree containing high-degree vertices and the lengths of paths attached to it, generalizing previous results on specific tree classes.

Contribution

It extends the understanding of how the chromatic symmetric function characterizes structural features of trees, including the size of the core and incident path lengths.

Findings

Chromatic symmetric function determines the size of the core subtree.

It also determines the multiset of incident path lengths.

Generalizes previous results on spider trees.

Abstract

For a tree $T$, consider its smallest subtree $T^{\circ}$ containing all vertices of degree at least $3$. Then the remaining edges of $T$ lie on disjoint paths each with one endpoint on $T^{\circ}$. We show that the chromatic symmetric function of $T$ determines the size of $T^{\circ}$, and the multiset of the lengths of these incident paths. In particular, this generalizes a proof of Martin, Morin, and Wagner that the chromatic symmetric function distinguishes spiders.