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

Abstract

A graph G is (du…,dl)-color able if the vertex set of G can be partitioned into subsets Vu…,Vl such that the graph G[Vi] induced by the vertices of Vi has maximum degree at most di for all 1 < i 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.