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2-Cell Embeddings with Prescribed Face Lengths and Genus
Bojan Mohar
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, B.C. V5A 1S6, Canada
Annals of Combinatorics 14 (4) pp.525-532 December, 2010
AMS Subject Classification: 05C07, 05C10
Let n be a positive integer, let d1,...,dn be a sequence of positive integers, and let q=1/2∑n{i=1} di. It is shown that there exists a connected graph G on n vertices, whose degree sequence is d1,...,dn and such that G admits a 2-cell embedding in every closed surface whose Euler characteristic is at least n-q+1, if and only if q is an integer and qn-1. Moreover, the graph G can be required to be loopless if and only if diq for i=1,...,n. This, in particular, answers a question of Skopenkov.
Keywords: degree sequence, 2-cell embedding, genus, maximum genus


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