q-Chromatic Polynomials

Algebra Seminar
Monday, April 1, 2024 - 1:00pm for 1 hour (actually 50 minutes)
Skiles 005
Andrés R. Vindas Meléndez – University of California, Berkeley
Changxin Ding
We introduce and study a $q$-version of the chromatic polynomial of a given graph $G=(V,E)$, namely,
\[\chi_G^\lambda(q,n) \ := \sum_{\substack{\text{proper colorings}\\ c\,:\,V\to[n]}} q^{ \sum_{ v \in V } \lambda_v c(v) },\] where $\lambda \in \mathbb{Z}^V$ is a fixed linear form.
Via work of Chapoton (2016) on $q$-Ehrhart polynomials, $\chi_G^\lambda(q,n)$ 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 $\chi_G^\lambda(q,n)$ 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.
This is joint work with Esme Bajo and Matthias Beck.