  DP-coloring (also called correspondence coloring) is a generalization of list
coloring that has been widely studied in recent years after its introduction by
Dvo\v{r}\'{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.
