Defective Coloring Revisited Lenore Cowen DEPARTMENT OF MATHEMATICAL SCIENCES AND DEPARTMENT OF COMPUTER SCIENCE JOHNS HOPKINS UNIVERSITY BALTIMORE, MARYLAND, USA Wayne Goddard DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF NATAL DURBAN, SOUTH AFRICA C. Esther Jesurum DEPARTMENT OF APPLIED MATHEMATICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASSACHUSETTS, USA ABSTRACT A graph is (k,d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2)- and (5, 1)-colorable, and that a graph of genus γ is (χ γ /(d + 1) + 4,d)-colorable, where χ γ is the maximum chromatic number of a graph embeddable on the surface of genus γ . It is shown that the (2,k)-coloring, for k ≥ 1, and the (3, 1)-coloring problems are NP-complete even for planar graphs. In general graphs (k,d)-coloring is NP-complete for k ≥ 3,d ≥ 0. The tightness is considered. Also, generalizations to defects of several algorithms for approximate (proper) coloring are presented. c  1997 John Wiley & Sons, Inc. 1. INTRODUCTION We define a (k,d)-coloring of a graph as a coloring of the vertices with k colors such that each vertex has at most d neighbors of its same color. For a graph G we define χ d (G) as the minimum k such that there is a (k,d)-coloring of G. So a (k, 0)-coloring is a proper coloring, and χ 0 (G) is the usual vertex chromatic number of the graph. Journal of Graph Theory Vol. 24, No. 3, 205 219 (1997) c  1997 John Wiley & Sons, Inc. CCC 0364-9024/97/030205-15