  DP-coloring is a generalization of list coloring that was introduced in 2015
by Dvo\v{r}\'{a}k and Postle. The chromatic polynomial of a graph is a notion
that has been extensively studied since the early 20th century. The chromatic
polynomial of graph $G$ is denoted $P(G,m)$, and it is equal to the number of
proper $m$-colorings of $G$. In 2019, Kaul and Mudrock introduced an analogue
of the chromatic polynomial for DP-coloring; specifically, the DP color
function of graph $G$ is denoted $P_{DP}(G,m)$. Two fundamental questions posed
by Kaul and Mudrock are: (1) For any graph $G$ with $n$ vertices, is it the
case that $P(G,m)-P_{DP}(G,m) = O(m^{n-3})$ as $m \rightarrow \infty$? and (2)
For every graph $G$, does there exist $p,N \in \mathbb{N}$ such that
$P_{DP}(K_p \vee G, m) = P(K_p \vee G, m)$ whenever $m \geq N$? We show that
the answer to both these questions is yes. In fact, we show the answer to (2)
is yes even if we require $p=1$.
