  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 $X_G$.
