Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\{ 4,5\}$-cycles and $d^{ \Delta}\ge d_0$ are $3$-choosable and planar graphs without $\{ 4,5,6\}$-cycles and $d^{ \Delta}\ge d_1$ are $3$-choosable, where $d^{ \Delta}$ is the smallest distance between triangles. They showed that $2\le d_0\le 4$ and $d_1\le 3$. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without $\{ 4,5\}$-cycles and $d^{ \Delta}\ge 3$ are DP-$3$-colorable, and (2) planar graphs without $\{ 4,5,6\}$-cycles and $d^{ \Delta}\ge 2$ are DP-$3$-colorable. DP-coloring is a generalization of list-coloring, thus as a corollary, $d_0\le 3$ and $d_1\le 2$. We actually prove stronger statements that each pre-coloring on some cycles can be extended to the whole graph.
Categories: math.CO