In 1979, Bernhart and Kainen conjectured that graphs of fixed genus g > 1 have unbounded pagenumber. In this paper. it is proven that genus g graphs can be embedded in 0(g) pages, thus disproving the conjecture. An 0( fi) lower bound is also derwed. The first algorithm in the literature for embedding an arbitra~graph in a book with a non-trlwal upper bound on the number of pages M presented. First, the algorithm computes the genus g of a graph using the algorithm of Filotti, Miller, Reif (

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... , which is polynomial-time for fixed genus. Second, it applies an optimal-time algorithm for obtaining an 0( g )-page book embedding. Separate book embedding algorithms are given for the cases of graphs embedded m orlentable and nonorientable surfaces. An important aspect of the construction is a new decomposition theorem, of independent interest, for a graph embedded on a surface. Book embedding has application in several areas, two of which are directly related to the results obtained: fault-tolerant VLSI and complexity theory, A preliminary announcement of this research appeared in