DP-4-colorability of planar graphs without adjacent cycles of given length

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle (2017). Kim and Ozeki proved that planar graphs without k-cycles where k=3,4,5, or 6 are DP-4-colorable. In this paper, we prove that every planar graph G without k-cycles adjacent to triangles is DP-4-colorable for k=5,6, which implies that every planar graph G without k-cycles adjacent to triangles is 4-choosable for k=5,6. This extends the result of Kim and Ozeki on 3-, 5-, and 6-cycles.