@MISC{Gebauer_onthe, author = {Heidi Gebauer}, title = {On the Number of Hamiltonian Cycles . . . }, year = {} }

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Abstract

The main contribution of this paper is a new approach for enumerating Hamilton cycles in bounded degree graphs – deriving thereby extremal bounds. We describe an algorithm which enumerates all Hamilton cycles of a given 3-regular n-vertex graph in time O(1.276 n), improving on Eppstein’s previous bound. The resulting new upper bound of O(1.276 n) for the maximum number of Hamilton cycles in 3-regular n-vertex graphs gets close to the best known lower bound of Ω(1.259 n). Our method differs from Eppstein’s in that he considers in each step a new graph and modifies it, while we fix (at the very beginning) one Hamilton cycle C and then proceed around C, succesively producing partial Hamilton cycles. Our approach can also be used to show that the number of Hamilton cycles of a 4-regular n-vertex graph is at most O(18 n/5) ≤ O(1.783 n), which improves a previous bound by Sharir and Welzl. This result is complemented by a lower bound of 48 n/8 ≥ 1.622 n. Then we present an algorithm which finds the minimum weight Hamilton cycle of a given 4-regular graph in time √ 3 n · poly(n) = O(1.733 n), improving a previous result of Eppstein. This algorithm can be modified to compute the number of Hamilton cycles in the same time bound and to enumerate all Hamilton cycles in time ( √ 3 n +hc(G))·poly(n) with hc(G) denoting the number of Hamilton cycles of the given graph G. So our upper bound of O(1.783 n) for the number of Hamilton cycles serves also as a time bound for enumeration. Using similar techniques as in the 3-regular case we establish upper bounds for the number of Hamilton cycles in 5-regular graphs and in graphs of average degree 3, 4, and 5.