Coloring with defect
- Johns Hopkins Univ., Baltimore, MD (United States)
- Univ. of Natal, Durban (South Africa)
- MIT, Cambridge, MA (United States)
An (ordinary vertex) coloring is a partition of the vertices of a graph into independent sets. The chromatic number is the minimum number of colors needed to produce such a partition. This paper considers a relaxation of coloring in which the color classes partition the vertices into subgraphs of degree at most d. d is called the defect of the coloring. A graph which admits a vertex coloring into k color classes, where each vertex is adjacent to at most d self-colored neighbors is said to be (k, d) colorable. We consider defective coloring on graphs of bounded degree, bounded genus, and bounded chromatic number, presenting complexity results and algorithms. For bounded degree graphs, a classic result of Lovasz yields a (k, [{Delta}/k]) coloring for graphs with E edges of maximum degree {Delta} in O({Delta}E) time. For graphs of bounded genus, (2, d), for d > 0 and (3,1)-coloring are proved NP-Complete, even for planar graphs. Results of easily can be transformed to (3,2) color any planar graph in linear time. We show that any toroidal graph is (3,2)- and (5, 1)-colorable, and quadratic-time algorithms are presented that find the colorings. For higher surfaces, we give a linear time algorithm to (3, {radical}12{gamma} + 6) color a graph of genus {gamma} > 2. It is also shown that any graph of genus {gamma} is ({radical}12{gamma}/(d + 1) + 6, d) colorable, and an O(d{radical}{gamma}E + V) algorithm is presented that finds the coloring. These bounds are within a constant factor of what is required for the maximum clique embeddable in the surface. Reductions from ordinary vertex coloring show that (k, d) coloring is NP-complete, and there exists an c > 0 such that no polynomial time algorithm can n{sup {epsilon}}-approximate the defective chromatic number unless P = NP. Most approximation algorithms to approximately color 3-colorable graphs can be extend to allow defects.
- OSTI ID:
- 471712
- Report Number(s):
- CONF-970142-; CNN: Grant N00014-96-1-0829; TRN: 97:001377-0061
- Resource Relation:
- Conference: 8. annual Association for Computing Machinery (ACM)-Society for Industrial and Applied Mathematics (SIAM) symposium on discrete algorithms, New Orleans, LA (United States), 5-7 Jan 1997; Other Information: PBD: 1997; Related Information: Is Part Of Proceedings of the eighth annual ACM-SIAM symposium on discrete algorithms; PB: 798 p.
- Country of Publication:
- United States
- Language:
- English
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