On the chromatic symmetric homology for star graphs

TL;DR

This paper proves and extends formulas for the chromatic symmetric homology of star graphs, a categorification of the chromatic symmetric function, demonstrating it as a stronger graph invariant.

Contribution

It advances the understanding of chromatic symmetric homology by proving and extending formulas specifically for star graphs, building on prior conjectures.

Findings

Proved new formulas for star graphs' chromatic symmetric homology.

Extended previous conjectures to a broader class of star graphs.

Confirmed that $H_*(G)$ is a stronger invariant than $X_G$.

Abstract

The chromatic symmetric function $X_G$ is a power series that encodes the proper colorings of a graph $G$ by assigning a variable to each color and a monomial to each coloring such that the power of a variable in a monomial is the number of times the corresponding color is used in the corresponding coloring. The chromatic symmetric homology $H_*(G)$ is a doubly graded family of $\mathbb{C}[\mathfrak{S}_n]$-modules that was defined by Sazdanovi\'c and Yip (2018) as a categorification of $X_G$. Chandler, Sazdanovi\'c, Stella, and Yip (2023) proved that $H_*(G)$ is a strictly stronger graph invariant than $X_G$, and they also computed or conjectured formulas for it in a number of special cases. We prove and extend some of their conjectured formulas for the case of star graphs, where one central vertex is connected to all other vertices and no other pairs of vertices are connected.