Geodesic
\(H\)-chromatic symmetric functions
The electronic journal of combinatorics, Tome 29 (2022) no. 1
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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.
DOI : 10.37236/10011
Classification : 05E05
Mots-clés : \(H\)-chromatic symmetric functions, proper \((H,\phi)\)-coloring, finitely distinguishable graph, uniformly distinguishable graph 
@article{10_37236_10011,
     author = {Nancy Mae Eagles and Ang\`ele M. Foley and Alice Huang and Elene Karangozishvili and Annan Yu},
     title = {\(H\)-chromatic symmetric functions},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {1},
     doi = {10.37236/10011},
     zbl = {1489.05155},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10011/}
}
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Nancy Mae Eagles; Angèle M. Foley; Alice Huang; Elene Karangozishvili; Annan Yu. \(H\)-chromatic symmetric functions. The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/10011

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