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We introduce and study a q-version of the chromatic polynomial of a given graph G= (V, E), namely, \ G^ (q, n) \: = ₑ₎₄ₑ ₂₎₋₎ₑ₈₍₆ₒ\\ ₂\,: \, ₕ[₍ q^ ₕ ₕ ᵥ c (v), \] where ZV is a fixed linear form. Via work of Chapoton (2016) on q-Ehrhart polynomials, G^ (q, n) turns out to be a polynomial in the q-integer nq, with coefficients that are rational functions in q. Additionally, we prove structural results for G^ (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.
Bajo et al. (Thu,) studied this question.