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An Introduction to Random Topological Graph Theory

Published online by Cambridge University Press:  12 September 2008

Arthur T. White
Arthur T. White
Affiliation:
Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008, USA
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Abstract

We introduce five probability models for random topological graph theory. For two of these models (I and II), the sample space consists of all labeled orientable 2-cell imbeddings of a fixed connected graph, and the interest centers upon the genus random variable. Exact results are presented for the expected value of this random variable for small-order complete graphs, for closed-end ladders, and for cobblestone paths. The expected genus of the complete graph is asymptotic to the maximum genus. For Model III, the sample space consists of all labeled 2-cell imbeddings (possibly nonorientable) of a fixed connected graph, and for Model IV the sample space consists of all such imbeddings with a rotation scheme also fixed. The event of interest is that the ambient surface is orientable. In both these models the complete graph is almost never orientably imbedded. The probability distribution in Models I and III is uniform; in Models II and IV it depends on a parameter p and is uniform precisely when p = 1/2. Model V combines the features of Models II and IV.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 4 , December 1994 , pp. 545 - 555
Copyright
Copyright © Cambridge University Press 1994

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