Crew, LoganHaithcock, EvanReynes, JosephineSpirkl, Sophie2024-05-242024-05-242024-07https://doi.org/10.1016/j.aam.2024.102718http://hdl.handle.net/10012/20593This is an open access article under the CC BY license (http://creativecommons.org /licenses/by /4.0/).In this paper, we extend the chromatic symmetric function X to a chromatic k-multisymmetric function Xk, 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 Xk. In particular, we show how to take advantage of homogeneous sets of G(those S⊆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 S1 S2 ⊆ V(G) generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs.enAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/chromatic symmetric functionmultisymmetric functionsymmetric functiondeletion-contractionstructural graph theoryStanley-Stembridge conjectureArticle