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Montassier, Raspaud, and Wang (2006) asked to find the smallest positiveintegers $d_0$ 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 $2\led_0\le 4$ and $d_1\le 3$. In this paper, we show that the following planargraphs 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 ageneralization of list-coloring, thus as a corollary, $d_0\le 3$ and $d_1\le2$. We actually prove stronger statements that each pre-coloring on some cyclescan be extended to the whole graph.

Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable