A deletion-contraction long exact sequence for chromatic symmetric homology

Crew and Spirklt generalize Stanley's chromatic symmetric function to vertex-weighted graphs. One of the primary motivations for extending the chromatic symmetric function to vertex-weighted graphs is the existence of a deletion-contraction relation in this setting, which, as known, holds for the chromatic polynomial, but doesn't hold for the chromatic symmetric function. 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 lifts the deletion-contraction relation that holds for the extension of Crew and Spirklt. Moreover, the new categorification gives a useful computational tool and allow us to answer two questions left open by Chandler, Sazdanovic, Stella and Yip. 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.