
An asymptotic combinatorial construction of 2Dsphere
A geometric space is constructed as the inverse limit of infinite sequen...
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A note on choosability with defect 1 of graphs on surfaces
This note proves that every graph of Euler genus μ is 2 + √(3μ + 3) c...
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Planar projections of graphs
We introduce and study a new graph representation where vertices are emb...
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On the dichromatic number of surfaces
In this paper, we give bounds on the dichromatic number χ⃗(Σ) of a surfa...
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Regular Polygon Surfaces
A regular polygon surface M is a surface graph (Σ, Γ) together with a co...
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Towards a short proof of the Fulek–Kynčl criterion for modulo 2 embeddability of graphs to surfaces
A connected graph K has a modulo 2 embedding to the sphere with g handle...
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Reappraising the distribution of the number of edge crossings of graphs on a sphere
Many real transportation and mobility networks have their vertices place...
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On the degree sequences of dual graphs on surfaces
Given two graphs G and G^* with a onetoone 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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