Stanley’s tree isomorphism problem

Determine whether the chromatic symmetric function distinguishes every pair of non-isomorphic trees.

Background

The paper introduces the chromatic symmetric function as an invariant that may encode structural information about a graph. It places its work in the context of Stanley’s tree isomorphism problem, which asks whether two non-isomorphic trees can have the same chromatic symmetric function. Although the invariant is known to distinguish several subclasses of trees and all trees up to 29 vertices, the general problem remains unresolved.

References

While this question remains open in general, the chromatic symmetric function has been shown to distinguish various subclasses of trees (see, e.g., ), as well as for all trees up to 29 vertices .

— The Chromatic Symmetric Function for Unicyclic Graphs  (2505.06486 - Bingham et al., 10 May 2025) in Section 1, Introduction

However, an extensively studied but still open question about $X_G$ is whether it is always different for different trees .

— Power sum expansions for Kromatic symmetric functions using Lyndon heaps  (2502.21285 - Pierson, 28 Feb 2025) in Section 1, subsection “Counting induced subgraphs using ${X}_G$”