q-Chromatic polynomials

We introduce and study a q-version of the chromatic polynomial of a given graph $G=(V,E)$, namely, \[ χ_G^λ(q,n) \ := ∑_{{proper colorings\\ c\,:\,V→[n]}} q^{ ∑v ∈ V λ_v c(v) }, \] where λ ∈ ZV is a fixed linear form. Via work of Chapoton (2016) on q-Ehrhart polynomials, χG^λ(q,n) turns out to be a polynomial in the q-integer [n]q, with coefficients that are rational functions in q. Additionally, we prove structural results for χG^λ(q,n) and exhibit connections to neighboring concepts, e.g., chromatic symmetric functions and the arithmetic of order polytopes. We offer a strengthened version of Stanley's conjecture that the chromatic symmetric function distinguishes trees, which leads to an analogue of P-partitions for graphs.