On chromatic symmetric homology and planarity of graphs (Q2684884)

Summary: \textit{R. Sazdanovic} and \textit{M. Yip} [J. Comb. Theory, Ser. A 154, 218--246 (2018; Zbl 1373.05204)] 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 \textit{A. Chandler} et al. [``On the strength of chromatic symmetric homology for graphs'', Preprint (2019; \url{arXiv:1911.13297})], 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.