  DP-coloring (also called correspondence coloring) is a generalization of list
coloring that was introduced by Dvo\v{r}\'{a}k and Postle in 2015. The
chromatic polynomial of a graph is an important notion in algebraic
combinatorics that was introduced by Birkhoff in 1912; denoted $P(G,m)$, it
equals the number of proper $m$-colorings of graph $G$. Counting function
analogues of chromatic polynomials have been introduced for list colorings:
$P_{\ell}$, list color functions (1990); DP colorings: $P_{DP}$, DP color
functions (2019), and $P^*_{DP}$, dual DP color functions (2021). For any graph
$G$ and $m \in \mathbb{N}$, $P_{DP}(G, m) \leq P_\ell(G,m) \leq P(G,m) \leq
P_{DP}^*(G,m)$. In 2022 (improving on older results) Dong and Zhang showed that
for any graph $G$, $P_{\ell}(G,m)=P(G,m)$ whenever $m \geq |E(G)|-1$.
Consequently, the list color function of a graph is a polynomial for
sufficiently large $m$. One of the most important and longstanding open
questions on DP color functions asks: for every graph $G$ is there an $N \in
\mathbb{N}$ and a polynomial $p(m)$ such that $P_{DP}(G,m) = p(m)$ whenever $m
\geq N$? We show that the answer to the analogue of this question for dual DP
color functions is no. Our proof reveals a connection between a dual DP color
function and the balanced chromatic polynomial of a signed graph introduced by
Zaslavsky in 1982.
