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Esther Tesar, Department of Mathematics, Drew University

**The genus distribution for a class of graphs called closed-end
ladders was
found by Furst, Gross and Statman. Their techniques can be extended
to
find probability polynomials (a generalization of the genus
distribution
of a cubic graph that instead of requiring each imbedding to be
equally
likely depends on a arbitrary probability p) for closed-end ladders. A
second extension that we make is using the genus distribution of
closed-end ladders to find the genus distribution of Ringel ladders.
We
then use this to find the average genus and to look at the probability
of finding an imbedding with one region used by Ringel and Youngs in
their proof of the Heawood Map Coloring Theorem.
**

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