Homogeneous sets in graphs and a chromatic multisymmetric function

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 C V ( G ) such that each vertex of V ( G ) \ 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 U S 2 C V ( G ) generalizes the process used by Guay-Paquet to reduce the StanleyStembridge conjecture to unit interval graphs. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).