On chromatic symmetric homology and planarity of graphs

TL;DR

This paper proves that non-planar graphs have a specific torsion in their chromatic symmetric homology, confirming a conjecture and linking graph planarity to algebraic properties of a categorified chromatic function.

Contribution

It establishes that non-planar graphs' chromatic symmetric homology contains Z_2-torsion, confirming a conjecture and connecting planarity with homological algebra.

Findings

Non-planar graphs have Z_2-torsion in their chromatic symmetric homology.

The proof uses a recursive argument based on Kuratowski's theorem.

Confirms the conjecture by Chandler, Sazdanovic, Stella, and Yip.

Abstract

Sazdanovic and Yip defined a categorification of Stanley's chromatic function called the chromatic symmetric homology. In this paper we prove that (as conjectured by Chandler, Sazdanovic, Stella and Yip), 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.