Chromatic Number

The chromatic number of a graph  is the smallest number of colors needed to color the vertices of  so that no two adjacent vertices share the same color (Skiena 1990, p. 210), i.e., the smallest value of  possible to obtain a k-coloring. Minimal colorings and chromatic numbers for a sample of graphs are illustrated above.

The chromatic number of a graph  is most commonly denoted  (e.g., Skiena 1990, West 2000, Godsil and Royle 2001, Pemmaraju and Skiena 2003), but occasionally also .

Empty graphs have chromatic number 1, while non-empty bipartite graphs have chromatic number 2.

The chromatic number of a graph  is also the smallest positive integer  such that the chromatic polynomial . Calculating the chromatic number of a graph is an NP-complete problem (Skiena 1990, pp. 211-212). Or, in the words of Harary (1994, p. 127), "no convenient method is known for determining the chromatic number of an arbitrary graph." However, Mehrotra and Trick (1996) devised a column generation algorithm for computing chromatic numbers and vertex colorings which solves most small to moderate-sized graph quickly.

Computation of the chromatic number of a graph is implemented in the Wolfram Language as VertexChromaticNumber[g]. Precomputed chromatic numbers for many named graphs can be obtained using GraphData[graph, "ChromaticNumber"].

The chromatic number of a graph must be greater than or equal to its clique number. A graph is called a perfect graph if, for each of its induced subgraphs , the chromatic number of  equals the largest number of pairwise adjacent vertices in . A graph for which the clique number is equal to the chromatic number (with no further restrictions on induced subgraphs) is said to be weakly perfect.

By definition, the edge chromatic number of a graph  equals the chromatic number of the line graph .

Brooks' theorem states that the chromatic number of a graph is at most the maximum vertex degree , unless the graph is complete or an odd cycle, in which case  colors are required.

A graph with chromatic number  is said to be bicolorable, and a graph with chromatic number  is said to be three-colorable. In general, a graph with chromatic number  is said to be an k-chromatic graph, and a graph with chromatic number  is said to be k-colorable.

The following table gives the chromatic numbers for some named classes of graphs.

graph
complete graph
cycle graph ,
star graph ,	2
wheel graph ,

For any two positive integers  and , there exists a graph of girth at least  and chromatic number at least  (Erdős 1961; Lovász 1968; Skiena 1990, p. 215).

The chromatic number of a surface of genus  is given by the Heawood conjecture,

where  is the floor function.  is sometimes also denoted  (which is unfortunate, since  commonly refers to the Euler characteristic). For , 1, ..., the first few values of  are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, ... (OEIS A000934).

Erdős (1959) proved that there are graphs with arbitrarily large girth and chromatic number (Bollobás and West 2000).

REFERENCES

Bollobás, B. and West, D. B. "A Note on Generalized Chromatic Number and Generalized Girth." Discr. Math. 213, 29-34, 2000.

Chartrand, G. "A Scheduling Problem: An Introduction to Chromatic Numbers." §9.2 in Introductory Graph Theory. New York: Dover, pp. 202-209, 1985.

Eppstein, D. "The Chromatic Number of the Plane." http://www.ics.uci.edu/~eppstein/junkyard/plane-color.html.Erdős, P. "Graph Theory and Probability." Canad. J. Math. 11, 34-38, 1959.

Erdős, P. "Graph Theory and Probability II." Canad. J. Math. 13, 346-352, 1961.

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 9, 1984.

Godsil, C. and Royle, G. Algebraic Graph Theory. New York: Springer-Verlag, 2001.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.

Lovász, L. "On Chromatic Number of Finite Set-Systems." Acta Math. Acad. Sci. Hungar. 19, 59-67, 1968.

Mehrotra, A. and Trick, M. A. "A Column Generation Approach for Graph Coloring." INFORMS J. on Computing 8, 344-354, 1996.

 https://mat.tepper.cmu.edu/trick/color.pdf.Pemmaraju, S. and Skiena, S. Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Cambridge, England: Cambridge University Press, 2003.

Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.

Sloane, N. J. A. Sequences A000012/M0003, A000934/M3292, A068917, A068918, and A068919 in "The On-Line Encyclopedia of Integer Sequences."Trick, West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2000.