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

The only Beraha numbers that occur as roots of chromatic polynomials are the integer values and (Harvey and Royle 2020). Royle (2016) constructed graphs having as a chromatic polynomial root, resolving the exceptional noninteger case.

The Beraha constants also arise as limit points of chromatic roots. Mactabi (2026) proved that every noninteger , namely or , is an accumulation point of real roots of the chromatic polynomials of simple plane triangulations. Mactabi (2026) reports Claude and ChatGPT assistance with the verification and audit scripts in Appendices A-C.

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References

Beraha, S. "Infinite Non-Trivial Families of Maps and Chromials." PhD thesis. Baltimore, MD: Johns Hopkins University, 1975.Harvey, D. J. and Royle, G. F. "Chromatic Roots at 2 and the Beraha Number

." J. Graph Th. 95, 445-456, 2020. https://doi.org/10.1002/jgt.22566.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 143, 1983.Mactabi, R. "Planar Disk Filters and Real Chromatic Roots at Beraha Numbers." 14 Oct 2026. https://arxiv.org/abs/2610.23989.Royle, G. "The Beraha Number

Is a Chromatic Root." 7 Nov 2016. https://arxiv.org/abs/1611.01941.Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, pp. 160-163, 1986.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. "Chromials." University of Waterloo, 1971.Tutte, W. T. "Chromatic Sums for Planar Triangulations I: The Case