The Kneser chromatic function distinguishes trees

TL;DR

This paper demonstrates that the Kneser chromatic function for k=2 uniquely identifies trees, advancing the understanding of graph invariants and their distinguishing power.

Contribution

It proves that the Kneser chromatic function with k=2 is a complete invariant for trees, extending previous results for k=1.

Findings

$X_{K_{ olinebreak ext{N},2}}$ uniquely identifies all trees.

The Kneser chromatic function with k=2 is a complete invariant for trees.

This extends Stanley's conjecture from k=1 to k=2.

Abstract

R.P. Stanley defined a invariant for graphs called the chromatic symmetric function and conjectured it is complete invariant for trees. Miezaki et al. generalised the chromatic symmetric function and defined the Kneser chromatic functions denoted by $\{X_{K_{\mathbb{N},k}}\}_{k\in\mathbb{N}}$, and rephrase Stanley's conjecture that $X_{K_{\mathbb{N},1}}$ is a complete invariant for trees. This paper shows $X_{K_{\mathbb{N},2}}$ is a complete invariant for trees.