DP-version of the Alon–Krivelevich–Sudakov conjecture

Let $G$ be a graph and let $\mathcal{H}=(L,H)$ be a DP-cover of $G$, meaning that $L$ assigns a color set $L(v)$ to each vertex and the edges between color sets corresponding to adjacent vertices form matchings. A proper $\mathcal{H}$-coloring chooses one color from each $L(v)$ so that the chosen colors form an independent set in $H$.

DP-version of the Alon–Krivelevich–Sudakov conjecture. For every graph $F$, there exist constants $c,d_0>0$ such that, whenever $H$ is $F$-free, has maximum degree $d\geqslant d_0$, and

$$|L(v)|\geqslant c d/\log d$$

for every $v\in V(G)$, the graph $G$ admits a proper $\mathcal{H}$-coloring.

This extends forbidden-subgraph coloring bounds from ordinary coloring to DP-coloring by imposing the forbidden-subgraph and degree conditions on the cover graph $H$. The source does not state a resolution, so the conjecture remains open.