2-Cell Embeddings with Prescribed Face Lengths and Genus

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.