On the strength of chromatic symmetric homology for graphs

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 over C as S_n-modules. We also show that integral chromatic symmetric homology contains torsion, and based on computations, conjecture that Z₂-torsion in bigrading (1,0) detects nonplanarity in the graph.