The.Algorithm.Design.Manual.Springer-Verlag.1998

Vertex Coloring
Notes: An excellent source on vertex coloring heuristics is Syslo, Deo, and Kowalik [SDK83], which
includes experimental results. Heuristics for vertex coloring include Brèlaz [Brè79], Matula [MMI72],
and Turner [Tur88]. Wilf [Wil84] proved that backtracking to test whether a random graph has chromatic
number k runs in constant time, dependent on k but independent of n. This is not as interesting as it
sounds, because only a vanishingly small fraction of such graphs are indeed k-colorable.
Expositions on algorithms to recognize bipartite graphs include [Man89]. Expositions on the hardness of
3-coloring graphs include [AHU74, Eve79a, Man89]. An interesting application of vertex coloring to
scheduling traffic lights appears in [AHU83].
Baase [Baa88] gives a very good description of approximation algorithms for graph coloring, including
Wigderson's [Wig83] factor of approximation algorithm, where is the chromatic
number of G. Hardness of approximation results for vertex coloring include [BGS95].
Brook's theorem states that the chromatic number , where is the maximum degree
of a vertex of G. Equality holds only for odd-length cycles (which have chromatic number 2) and
complete graphs.
The most famous problem in the history of graph theory is the four-color problem, first posed in 1852
and finally settled in 1976 by Appel and Haken using a proof involving extensive computation. Any
planar graph can be 5-colored using a variation of the color interchange heuristic. Despite the four-color
theorem, it is NP-complete to test whether a particular planar graph requires four colors or whether three
suffice. See [SK86] for an exposition on the history of the four-color problem and the proof. An efficient
algorithm to four-color a graph is presented in [RSST96].
Related Problems: Independent set (see page ), edge coloring (see page ).
Next: Edge Coloring Up: Graph Problems: Hard Problems Previous: Graph Partition
Algorithms
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