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Let G be a graph without 4-cycles and 5-cycles. We show that the problem to determine whether G is (0, k)-colorable is NP-complete for each positive integer k. Moreover, we construct non-(1, k)-colorable planar graphs without 4-cycles and 5-cycles for each positive integer k. Finally, we prove that G is (d 1 , d 2 )-colorable where (d 1 , d 2 ) = (4, 4), (3, 5), and (2,9).

In [7]), Borodin and Ivanova proved that every planar graph without 4-cycles adjacent to 3cycles is list vertex 2-aborable. In fact, they proved a more general result in terms of variable degeneracy. Inspired by these results and DP-coloring which becomes a widely studied topic, we introduce a generalization on variable degeneracy including list vertex arboricity. We use this notion to extend a general result by Borodin and Ivanova. Not only that this theorem implies results about planar graphs without 4-cycles adjacent to 3-cycle by Borodin and Ivanova, it also implies many other results including a result by Kim and Yu [20] that every planar graph without 4-cycles adjacent to 3-cycles is DP-4-colorable.

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