Proper $q$-caterpillars are distinguished by their Chromatic Symmetric Functions

TL;DR

This paper proves Stanley's Tree Isomorphism Conjecture for a new class of trees called proper $q$-caterpillars, extending the class of trees distinguished by their chromatic symmetric functions.

Contribution

The paper introduces proper $q$-caterpillars and proves that their chromatic symmetric functions distinguish non-isomorphic trees, broadening the scope of Stanley's conjecture.

Findings

Proper $q$-caterpillars are distinguished by their chromatic symmetric functions.

The conjecture holds for this new class of trees.

This extends previous results on caterpillars and related subclasses.

Abstract

Stanley's Tree Isomorphism Conjecture posits that the chromatic symmetric function can distinguish non-isomorphic trees. While already established for caterpillars and other subclasses of trees, we prove the conjecture's validity for a new class of trees that generalize proper caterpillars, thus confirming the conjecture for a broader class of trees.