On the Strength of Chromatic Symmetric Homology for graphs

TL;DR

This paper explores the properties of chromatic symmetric homology as a graph invariant, revealing its ability to distinguish graphs beyond the chromatic symmetric function and uncovering torsion phenomena related to graph planarity.

Contribution

It demonstrates that chromatic symmetric homology can differentiate graphs with identical chromatic symmetric functions and introduces the conjecture linking torsion to nonplanarity.

Findings

Identified pairs of graphs with same chromatic symmetric function but different homology.

Proved that integral chromatic symmetric homology contains torsion.

Conjectured that Z_2-torsion detects nonplanarity in graphs.

Abstract

In this paper, we investigate the strength of chromatic symmetric homology as a graph invariant. Chromatic symmetric homology is a lift of the chromatic symmetric function for graphs to a homological setting, and its Frobenius characteristic is a q,t generalization of the chromatic symmetric function. We exhibit three pairs of graphs where each pair has the same chromatic symmetric function but distinct homology. We also show that integral chromatic symmetric homology contains torsion, and based on computations, conjecture that Z_2-torsion in bigrading (1,0) detects nonplanarity in the graph.