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A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitionedinto subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices ofV_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focuson complexity aspects of such colorings when l=2,3. More precisely, we provethat, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3,either every planar graph with girth at least g is (k,j)-colorable or it isNP-complete to determine whether a planar graph with girth at least g is(k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determinewhether a planar graph that is either (0,0,0)-colorable ornon-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibitnon-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planargraphs with girth 7.

Near-colorings: non-colorable graphs and NP-completeness