2-Cell Embeddings with Prescribed Face Lengths and Genus

Abstract

Let n be a positive integer, let d1, . . . , dn 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 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 q ≥ n − 1. Moreover, the graph G can be required to be loopless if and only if di ≤ q for i = 1, . . . , n. This, in particular, answers a question of Skopenkov.