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Chromatic Sums for Rooted Planar Triangulations: The Cases λ = 1 and λ = 2

Summary In this paper we derive a functional equation whose solution would give the sum of the chromatic polynomial P(M, λ) over certain classes of rooted planar maps M called “ triangulations” and “near-triangulations”. For an integral colour-number λ this sum is the number of λ-coloured rooted maps of the kind considered, but the sum can also be discussed for non-integral λ.