On the chromatic symmetric functions of trees

Abstract

The chromatic symmetric function (CSF) of a graph, introduced by Richard Stanley in 1995, is a symmetric function generalization of the chromatic polynomial which enumerates proper colorings by the number of uses of each color. We attack a long-standing open question of Stanley that asks whether trees are distinguished by their CSFs. We prove new structural results about the CSFs of well-known subclasses of trees, including an involution on symmetric functions that swaps the CSFs of caterpillars in pairs, as well as a characterization of all linear relations between the CSFs of spiders. Then, we study a related graph invariant, the generalized degree polynomial (GDP), introduced by Crew and shown to be determined by the CSF by Aliste-Prieto et al. We present several classes of data about a tree that can be recovered from its GDP (and thus also from its CSF), such as the double-degree sequence and leaf adjacency sequence; this extends previous work of Martin, Morin, and Wagner. Then, we consider vector-valued and matrix-valued variants of the GDP for trees with distinguished vertices. By using linear relations that these GDPs satisfy, we prove new recurrence relations for the ordinary GDP and give several constructions that produce families of ordinary trees with the same GDP. In addition, we prove in some cases that the trees produced by such constructions have different CSFs. Our work suggests a program for further work using the GDP as an intermediary.