Spanning trees and continued fractions

Abstract

We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple (and planar) graphs on $n$ vertices, answering a question of Sedláček from 1969. The proof uses a connection with continued fractions, ``thin orbits,'' and Zaremba's conjecture.

21 pages, 7 figures. New references are added to v2