Defective 2-colorings of planar graphs without 4-cycles and 5-cycles

A 2-coloring is a coloring of vertices of a graph with colors 1 and 2. Define V-i := {v is an element of V(G) : c(v) = i} for i = 1 and 2. We say that G is (d(1), d(2))-colorable if G has a 2-coloring such that V-i is an empty set or the induced subgraphG[V-i] has the maximum degree at most d(i) for i = 1 and 2. Let G be a planar graph without 4-cycles and 5-cycles. We show that the problem to determine whether G is (0, k)-colorable is NP-complete for every positive integer k. Moreover, we construct non -(1, k)-colorable planar graphs without 4-cycles and 5-cycles for every positive integer k. In contrast, we prove that G is (d(1), d(2))-colorable where (d(1), d(2)) = (4, 4), (3, 5), and (2, 9). (C) 2018 Elsevier B.V. All rights reserved.