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Chromatic Sums for Rooted Planar Triangulations II: The Case λ = τ + 1

Published online by Cambridge University Press:  20 November 2018

W. T. Tutte
W. T. Tutte*
Affiliation:
University of Waterloo, Waterloo, Ontario
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Summary

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In an earlier paper [2] we denned the chromatic sums g, q, l and h. We determined the derivatives of these sums with respect to the colour-number λ at the special values λ = 1 and λ = 2. In the present paper we find parametric equations for h, l and q in the case λ = τ + 1, where τ is the golden ratio. We obtain h, l and the basic indeterminate z explicitly in terms of the parameter u, but for q we exhibit only a cubic equation with coefficients depending on u. We obtain an exact formula for the coefficients in h by applying Lagrange's theorem to the parametric equations.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 3 , June 1973 , pp. 657 - 671
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Birkhoff, G. D. and Lewis, D. C., Chromatic polynomials, Trans. Amer. Math. Soc. 60 (1946), 355–451.Google Scholar
2. Tutte, W. T., Chromatic sums for planar triangulations: the cases X = 1 and X = 2, Can. J. Math. 25 (1973), 426–447.Google Scholar
3. Tutte, W. T., On chromatic polynomials and the golden ratio, J. Combinatorial Theory 9 (1970), 289–296.Google Scholar
4. Tutte, W. T., The golden ratio in the theory of chromatic polynomials, Annals of the New York Academy of Sciences 175 (1970), 391–402.Google Scholar