Abstract
The equations of the title appear in the author's paper “Chromatic Sums for Rooted Planar Triangulations, V: Special Equations.” (Canadian Journal of Mathematics, 26 (1974), 893–907). They appear in that paper as Equations (24) and (25). They are simultaneous equations for two unknown functionsl andy 2 of two variablesy 1 andz. A parameterμ is involved. The main result is that forμ = 2 cos (2π/n), wheren is a positive integer >1, the two equations can be reduced to a single equation (numbered (49)). Solutions of this are known forn <7. From such solutions we can expect to get information about the averaged chromatic polynomials of planar triangulations with a given number of triangles.
The present work is basically an expository paper on the theory given in “Chromatic Sums, V,” but it includes some new results and many simplifications.
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References
Tutte, W. T.,Chromatic sums for rooted planar triangulations: the cases λ = 1 and λ = 2. Can. J. Math.25 (1973), 426–447.
Tutte, W. T.,Chromatic sums for rooted planar triangulations II: the case λ = τ + 1. Can. J. Math.25 (1973), 657–671.
Tutte, W. T.,Chromatic sums for rooted planar triangulations III: the case λ = 3. Can. J. Math.25 (1973), 780–790.
Tutte, W. T.,Chromatic sums for rooted planar triangulations V: special equations. Can. J. Math.26 (1974), 893–907.
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Tutte, W.T. On a pair of functional equations of combinatorial interest. Aequat. Math. 17, 121–140 (1978). https://doi.org/10.1007/BF01818554
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DOI: https://doi.org/10.1007/BF01818554