A deletion–contraction long exact sequence for chromatic symmetric homology

In Crew and Spirkl (2020), the authors generalize Stanley’s chromatic symmetric function (Stanley, 1995) to vertex-weighted graphs. In this paper we find a categorification of their new invariant extending the definition of chromatic symmetric homology to vertex-weighted graphs. We prove the existence of a deletion–contraction long exact sequence for chromatic symmetric homology which gives a useful computational tool and allow us to answer two questions left open in Chandler et al. (2019). In particular, we prove that, for a graph G with n vertices, the maximal index with nonzero homology is not greater that n-1. Moreover, we show that the homology is non-trivial for all the indices between the minimum and the maximum with this property.