Homogeneous sets in graphs and a chromatic multisymmetric function
Abstract
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.
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Cite this version of the work
Logan Crew, Evan Haithcock, Josephine Reynes, Sophie Spirkl
(2024).
Homogeneous sets in graphs and a chromatic multisymmetric function. UWSpace.
http://hdl.handle.net/10012/20593
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