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On the genus of a graph
by
Liangxia Wan
Beijing Jiaotong University
Determining the genus of a graph can be dated to the Heawood Conjecture in 1890. The Conjecture is implied by the Thread Problem by Hilbert and Cohn-Vossen which is the genus problem of a complete graph. A current graph was introduced by Gustin and a generalized current graph was done by Youngs in 1963. Ringel classified the genera of complete graphs and then this problem was solved in 1968.
Later, researchers mainly studied the genera of other special graphs with certain symmetry which are complete bipartite graphs， complete multipartite graphs n -cube etc. Until now the genera of complete tripartite graphs are not yet fully determined. In fact, Thomassen has proved that determining the genus of a graph( even a cubic graph) is NP-complete.
In this talk a formal set and its accompanying graph are introduced and a planarity rule of a formal set is established. The planarity rule is used to determine the genera of more general graphs. As an example, genera of a type of 3-connected cubic graphs are provided. In addition, the genus of a graph can be applied in field of moduli spaces of curves.
Date received: November 25, 2015
Copyright © 2015 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cblw-42.