A combinatorial analog of the Jordan Curve Theorem

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Abstract

The concept of the genus of a pair of permutations is defined in the same manner as was done by Jacques. The integrality of the genus is proven in a new way by applying a technique developed by Walkup for the reduction of products of permutations. These tools are then used to prove an analog of the Jordan Curve Theorem for a pair of permutations whose genus is zero. A Jordan Curve Theorem for plane embeddings of graphs then follows as a corollary.

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This investigation was supported by University of Kansas General Research Allocation No. 3118-X0-0038.