Beraha Constants

The th Beraha constant (or number) is given by

 is , where  is the golden ratio,  is the silver constant, and . The following table summarizes the first few Beraha numbers.

		approx.
1	4
2	0
3	1
4	2
5		2.618
6	3
7		3.247
8		3.414
9		3.532
10		3.618

Noninteger Beraha numbers can never be roots of any chromatic polynomials with the possible exception of  (G. Royle, pers. comm., Nov. 21, 2005). However, the roots of chromatic polynomials of planar triangulations appear to cluster around the Beraha numbers (and, technically, are conjectured to be accumulation points of roots of planar triangulation chromatic polynomials).

REFERENCES:

Beraha, S. Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 143, 1983.

Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, pp. 160-163, 1986.

Tutte, W. T. "Chromials." University of Waterloo, 1971.

Tutte, W. T. "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.

Tutte, W. T. "Chromatic Sums for Planar Triangulations I: The Case ." Research Report COPR 72-7, University of Waterloo, 1972a.

Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case ." Research Report COPR 72-4, University of Waterloo, 1972b.