Non-chromatic-Adherence of the DP Color Function via Generalized Theta Graphs

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvorak and Postle in 2015. The chromatic polynomial of a graph is an extensively studied notion in combinatorics since its introduction by Birkhoff in 1912; denoted P(G, m), it equals the number of proper m-colorings of graph G. Counting function analogues of the chromatic polynomial have been introduced and studied for list colorings: Pt, the list color function (1990); DP colorings: PDP, the DP color function (2019), and P-DP(*), the dual DP color function (2021). For any graph G and m ? N, P-DP(G, m) = P-l(G, m) = P(G, m) = P-DP(*) (G, m). A function f is chromatic-adherent if for every graph G, f (G, a) = P(G, a) for some a = ?(G) implies that f (G, m) = P(G, m) for all m = a. It is not known if the list color function and the DP color function are chromatic-adherent. We show that the DP color function is not chromatic-adherent by studying the DP color function of Generalized Theta graphs. The tools we develop along with the Rearrangement Inequality give a new method for determining the DP color function of all Theta graphs and the dual DP color function of all Generalized Theta graphs.