Voltage graphs

A possible way to obtain a complicated graph imbedding in a surface is to derive it as a covering of a simpler imbedding by assigning ''voltages'' to the edges of the simpler imbedding. Although this new method is but the dual of the current graph construction,it has independent importance in practical applications, illustrated here by a computation of the genus of a class of metacyclic groups.

[1]  Jonathan L. Gross,et al.  Quotients of complete graphs: revisiting the Heawood map-coloring problem. , 1974 .

[2]  Arthur T. White.  Orientable imbeddings of Cayley graphs , 1974 .

[3]  A. Tucker,et al.  Branched and folded coverings , 1936 .

[4]  Mark Jungerman.  The genus of the symmetric quadripartite graph , 1975 .

[5]  A. White.  On the genus of a group , 1972 .

[6]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[7]  J. Gross,et al.  The topological theory of current graphs , 1974 .

[8]  L. W. Beineke.  Minimal decompositions of complete graphs into subgraphs with embeddability properties , 1969 .

[9]  H. Zassenhaus.  The theory of groups , 1949 .

[10]  I. Anderson,et al.  Bi-embeddings of graphs , 1974, Glasgow Mathematical Journal.

[11]  H. Maschke.  The Representation of Finite Groups, Especially of the Rotation Groups of the Regular Bodies of Three-and Four-Dimensional Space, by Cayley's Color Diagrams , 1896 .

[12]  F. Baebler.  Über die Zerlegung regulärer Streckenkomplexe ungerader Ordnung , 1937 .

[13]  G. Ringel,et al.  Die toroidale Dicke des vollständigen Graphen , 1965 .

[14]  A. Clifford,et al.  The algebraic theory of semigroups , 1964 .

[15]  A. White.  The genus of the complete tripartite graph Kmn,n,n , 1969 .

[16]  Seth R. Alpert,et al.  Twofold Triple Systems and Graph Imbeddings , 1975, J. Comb. Theory, Ser. A.

[17]  William Gustin,et al.  Orientable embedding of Cayley graphs , 1963 .

[18]  Jonathan L. Gross,et al.  Branched coverings of graph imbeddings , 1973 .

[19]  G. Ringel,et al.  Das Geschlecht des vollständigen dreifärbbaren Graphen , 1970 .

[20]  G. Ringel,et al.  Solution of the heawood map-coloring problem. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[21]  L. Heffter.  Ueber das Problem der Nachbargebiete , 1891 .

[22]  J. W. Alexander.  Note on Riemann spaces , 1920 .