A Deletion-Contraction Relation for the Chromatic Symmetric Function

TL;DR

This paper introduces a deletion-contraction relation for the chromatic symmetric function of vertex-weighted graphs, enabling new proofs and properties to be derived, thus advancing understanding of this graph invariant.

Contribution

It extends the chromatic symmetric function to weighted graphs and establishes a deletion-contraction relation similar to the chromatic polynomial, providing new insights and proofs.

Findings

Derived a deletion-contraction relation for weighted graphs

Provided new properties of the chromatic symmetric function

Supplied alternative proofs for fundamental properties

Abstract

We extend the definition of the chromatic symmetric function $X_G$ to include graphs $G$ with a vertex-weight function $w : V(G) \rightarrow \mathbb{N}$. We show how this provides the chromatic symmetric function with a natural deletion-contraction relation analogous to that of the chromatic polynomial. Using this relation we derive new properties of the chromatic symmetric function, and we give alternate proofs of many fundamental properties of $X_G$.