On the Number of Hamilton Cycles in Bounded Degree Graphs

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(1.276n), improving on Eppstein's previous bound. The resulting new upper bound of O(1.276n) for the maximum number of Hamilton cycles in 3-regular n-vertex graphs gets close to the best known lower bound of Ω(1.259n). 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(18n/5) ≤ O(1.783n), which improves a previous bound by Sharir and Welzl. This result is complemented by a lower bound of 48n/8 ≥ 1.622n. Then we present an algorithm which finds the minimum weight Hamilton cycle of a given 4-regular graph in time √3n · poly(n) = O(1.733n), 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 (√3n +hc(G))·poly(n) with hc(G) denoting the number of Hamilton cycles of the given graph G. So our upper bound of O(1.783n) 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.