On the degree sequences of dual graphs on surfaces
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Given two graphs G and G^* with a one-to-one correspondence between their edges, when do G and G^* form a pair of dual graphs realizing the vertices and countries of a map embedded in a surface? A criterion was obtained by Jack Edmonds in 1965. Furthermore, let d=(d_1,…,d_n) and t=(t_1,…,t_m) be their degree sequences. Then, clearly, ∑_i=1^n d_i = ∑_j=1^m t_j = 2ℓ, where ℓ is the number of edges in each of the two graphs, and χ = n - ℓ + m is the Euler characteristic of the surface. Which sequences d and t satisfying these conditions still cannot be realized as the degree sequences? We make use of Edmonds' criterion to obtain several infinite series of exceptions for the sphere, χ = 2, and projective plane, χ = 1. We conjecture that there exist no exceptions for χ≤ 0.
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