  In 1980, Albertson and Berman introduced partial coloring. In 2000,
Albertson, Grossman, and Haas introduced partial list coloring. Here, we
initiate the study of partial coloring for an insightful generalization of list
coloring introduced in 2015 by Dvo\v{r}\'{a}k and Postle, DP-coloring (or
correspondence coloring). We consider the DP-coloring analogue of the Partial
List Coloring Conjecture, which generalizes a natural bound for partial
coloring. We show that while this partial DP-coloring conjecture does not hold,
several results on partial list coloring can be extended to partial
DP-coloring. We also study partial DP-coloring of the join of a graph with a
complete graph, and we present several interesting open questions.
