Planar graphs without mutually adjacent 3-, 5-, and 6-cycles are 3-degenerate

A graph G is k-degenerate if every subgraph of G has a vertex with degree at most k. Using the Euler's formula, one can obtain that planar graphs without 3-cycles are 3-degenerate. Wang and Lih, and Fijavž et al. proved the analogue results for planar graphs without 5-cycles and planar graphs without 6-cycles, respectively. Recently, Liu et al. showed that planar graphs without 3-cycles adjacent to 5-cycles are 3-degenerate. In this work, we generalized all aforementioned results by showing that planar graphs without mutually adjacent 3-,5-, and 6-cycles are 3-degenerate. A graph G without mutually adjacent 3-,5-, and 6-cycles means that G cannot contain three graphs, say G1,G2, and G3, where G1 is a 3-cycle, G2 is a 5-cycle, and G3 is a 6-cycle such that each pair of G1,G2, and G3 are adjacent. As an immediate consequence, we have that every planar graph without mutually adjacent 3-,5-, and 6-cycles is DP-4-colorable. This consequence also generalizes the result by Chen et al that planar graphs without 5-cycles adjacent to 6-cycles are DP-4-colorable.