In this article, we use a unified approach to prove several classes of planar graphs are DP-3colorable, which extend the corresponding results on 3-choosability.

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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 ∆ ≥ d 0 are 3-choosable and planar graphs without {4, 5, 6}-cycles and d ∆ ≥ d 1 are 3-choosable, where d ∆ is the smallest distance between triangles. They showed that 2 ≤ d 0 ≤ 4 and d 1 ≤ 3. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without {4, 5}-cycles and d ∆ ≥ 3 are DP-3-colorable, and (2) planar graphs without {4, 5, 6}-cycles and d ∆ ≥ 2 are DP-3-colorable. DP-coloring is a generalization of list-coloring, thus as a corollary, d 0 ≤ 3 and d 1 ≤ 2. We actually prove stronger statements that each pre-coloring on some cycles can be extended to the whole graph.

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