A Note on Distinguishing Trees with the Chromatic Symmetric Function

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.