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Whereas the topological theory of covering spaces describes an existential relationship between the domain and the codomain of a mapping, the theory of voltage graphs, due to Gross [7] and Gross and Tucker [12], provides a combinatorial tool for constructing graphs and graph embeddings. In voltage graph theory, the many specialized forms of combinatorial current graph originating with Gustin and augmented by Ringel and Youngs (see [29]) are all unified, so that the Ringel–Youngs embeddings are readily understood as the duals of coverings of voltage graphs (see [9] and [11]).

E. A. Nordhaus, B. M. Stewart and A. T. White, On the maximum genus of a graph, J. Combin. Theory (B) 11 (1971), 258–267. 24. C. D. Papakyriakopoulos, A new proof of the invariance of the homology groups of a complex, Bull. Soc. Math. Grèce 22 (1943), 1–154. 25. V. K. Proulx, Classification of the toroidal groups, J. Graph Theory 2 (1981), 269–273. 26. M. O. Rabin, Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958), 172–194. 27. T. Rado, Über den Begriff der Riemannschen Flache, Acta Litt.

Gross and Thomas W. Tucker with a variety of standard enumerative methods. Such inventories are the topic of Chapter 3. In recent years, Kwak and Lee have led in the application of voltage graph methods for enumerating graph coverings, and Chapter 9 provides an account of this active branch of topological graph theory. Combinatorial methods predominated in the older, complementary programme of research launched by Tutte [40], [41] into the counting of maps on a given surface. Jackson and Visentin [19] have provided a complete listing of the maps with a small number of edges.

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