Coloring with Defects

This paper is concerned with algorithms and complexity results for defective coloring, where a defective (k; d)-coloring is a k coloring of the vertices of a graph such that each vertex is adjacent to at most d-self-colored neighbors. First, (2; d) coloring is shown NP-complete for d 1, even for planar graphs, and (3; 1) coloring is also shown NP-complete for planar graphs (while there exists a quadratic algorithm to (3,2)-color any planar graph). A reduction from ordinary vertex coloring then shows ( ; d) coloring NP-complete for any 3; d 0, as well as hardness of approximation results. Second, a generalization of + 1 coloring to defects is explored for graphs of maximum degree . Based on a theorem of Lovasz, we obtain an O(E) algorithm to (k; b k c) color any graph; this yields an O(E) algorithm to (2,1)-color 3-regular graphs, and (3,2)-color 6-regular graphs. The generalization of + 1 coloring is used in turn to generalize the polynomial-time approximate 3and k-coloring algorithms of Widgerson and Karger-Motwani-Sudan to allow defects. For approximate 3-coloring, we obtain an O( E) time algorithm to (d(8n d ):5e; d) color, and a polynomial time algorithm to (O((nd ):387); d) color any 3-colorable graph.