$H$-Chromatic Symmetric Functions
-
Published:2022-02-11
Issue:1
Volume:29
Page:
-
ISSN:1077-8926
-
Container-title:The Electronic Journal of Combinatorics
-
language:
-
Short-container-title:Electron. J. Combin.
Author:
Eagles Nancy Mae,Foley Angèle M.,Huang Alice,Karangozishvili Elene,Yu Annan
Abstract
We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementary symmetric functions, can be realized as $H$-chromatic symmetric functions. Moreover, we show that if $G$ and $H$ are particular types of multipartite complete graphs we can derive a set of $H$-chromatic symmetric functions that are a basis for $\Lambda^n$. We end with some conjectures and open problems.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics