  The DP-coloring is a generalization of the list coloring, introduced by
Dvo\v{r}\'{a}k and Postle. Let $\mathcal{H}=(L,H)$ be a cover of a graph $G$
and $P_{DP}(G,\mathcal{H})$ be the number of $\mathcal{H}$-colorings of $G$.
The DP color function $P_{DP}(G,m)$ of $G$, introduced by Kaul and Mudrock, is
the minimum value of $P_{DP}(G,\mathcal{H})$ where the minimum is taken over
all possible $m$-fold covers $\mathcal{H}$ of $G$. For the family of $n$-vertex
connected graphs, one can deduce that trees maximize the DP color function,
from two results of Kaul and Mudrock. In this paper we obtain tight upper
bounds for the DP color function of $n$-vertex $2$-connected graphs. Another
concern in this paper is the canonical labeling in a cover. It is well known
that if an $m$-fold cover $\mathcal{H}$ of a graph $G$ has a canonical
labeling, then $P_{DP}(G,\mathcal{H})=P(G,m)$ in which $P(G,m)$ is the
chromatic polynomial of $G$. However the converse statement of this conclusion
is not always true. We give examples that for some $m$ and $G$, there exists an
$m$-fold cover $\mathcal{H}$ of $G$ such that $P_{DP}(G,\mathcal{H})=P(G,m)$,
but $\mathcal{H}$ has no canonical labelings. We also prove that when $G$ is a
unicyclic graph or a theta graph, for each $m\geq 3$, if
$P_{DP}(G,\mathcal{H})=P(G,m)$, then $\mathcal{H}$ has a canonical labeling.
