Abstract
Let n be a positive integer, let d 1, . . . , d n be a sequence of positive integers, and let \({{q = \frac{1}{2}\sum^{n}_{i=1} d_{i}\cdot}}\). It is shown that there exists a connected graph G on n vertices, whose degree sequence is d 1, . . . , d n 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 q ≥ n − 1. Moreover, the graph G can be required to be loopless if and only if d i ≤ q for i = 1, . . . , n. This, in particular, answers a question of Skopenkov.
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Supported in part by the ARRS, Research Program P1–0507, and by an NSERC Discovery Grant.
On leave IMFM & FMF, Department of Mathematics, University of Ljubljana, 1000 Ljubljana, Slovenia.
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Mohar, B. 2-Cell Embeddings with Prescribed Face Lengths and Genus. Ann. Comb. 14, 525–532 (2010). https://doi.org/10.1007/s00026-011-0075-8
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DOI: https://doi.org/10.1007/s00026-011-0075-8