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 Dvoř\'{a}k 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: \(P_{\ell}\), the list color function (1990); DP colorings: \(P_{DP}\), the DP color function (2019), and \(P^*_{DP}\), the dual DP color function (2021). For any graph \(G\) and \(m \in \mathbb{N}\), \(P_{DP}(G, m) \leq P_\ell(G,m) \leq P(G,m) \leq P_{DP}^*(G,m)\). A function \(f\) is chromatic-adherent if for every graph \(G\), \(f(G,a) = P(G,a)\) for some \(a \geq \chi(G)\) implies that \(f(G,m) = P(G,m)\) for all \(m \geq a\). 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.