Near-Colorings: Non-Colorable Graphs and NP-Completeness

A graph G is (d1, . . . , dl)-colorable if the vertex set of G can be partitioned into subsets V1, . . . , Vl such that the graph G[Vi] induced by the vertices of Vi has maximum degree at most di for all 1 6 i 6 l. In this paper, we focus on complexity aspects of such colorings when l = 2, 3. More precisely, we prove that, for any fixed integers k, j, g with (k, j) 6= (0, 0) and g > 3, either every planar graph with girth at least g is (k, j)-colorable or it is NP-complete to determine whether a planar graph with girth at least g is (k, j)-colorable. Also, for every fixed integer k, it is NP-complete to determine whether a planar graph that is either (0, 0, 0)-colorable or non-(k, k, 1)-colorable is (0, 0, 0)-colorable. Additionally, we exhibit non-(3, 1)colorable planar graphs with girth 5 and non-(2, 0)-colorable planar graphs with girth 7.