On Chromatic Symmetric Homology and Planarity of Graphs

Abstract

Sazdanovic and Yip (2018) defined a categorification of Stanley’s chromatic symmetric function called the chromatic symmetric homology, given by a suitable family of representations of the symmetric group. In this paper we prove that, as conjectured by Chandler, Sazdanovic, Stella and Yip (2019), if a graph $G$ is non-planar, then its chromatic symmetric homology in bidegree (1,0) contains $\mathbb{Z}_2$-torsion. Our proof follows a recursive argument based on Kuratowsky’s theorem.