Homogeneous Sets in Graphs and a Chromatic Multisymmetric Function

TL;DR

This paper introduces a chromatic multisymmetric function for graphs with vertex partitions, enabling new linear relations and simplifying the analysis of homogeneous sets to address the Stanley-Stembridge conjecture.

Contribution

It extends the chromatic symmetric function to a multisymmetric version for graphs with vertex partitions, providing a systematic way to derive relations and analyze homogeneous sets.

Findings

Defined the chromatic $k$-multisymmetric function $X_k$

Established properties and basis expansions of $X_k$

Applied the method to reduce the Stanley-Stembridge conjecture

Abstract

In this paper, we extend the chromatic symmetric function $X$ to a chromatic $k$-multisymmetric function $X_k$, defined for graphs equipped with a partition of their vertex set into $k$ parts. We demonstrate that this new function retains the basic properties and basis expansions of $X$, and we give a method for systematically deriving new linear relationships for $X$ from previous ones by passing them through $X_k$. In particular, we show how to take advantage of homogeneous sets of $G$ (those $S \subseteq V(G)$ such that each vertex of $V(G) \backslash S$ is either adjacent to all of $S$ or is nonadjacent to all of $S$) to relate the chromatic symmetric function of $G$ to those of simpler graphs. Furthermore, we show how extending this idea to homogeneous pairs $S_1 \sqcup S_2 \subseteq V(G)$ generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs.