Finding and enumerating Hamilton cycles in 4-regular graphs

Abstract

We show that each 4-regular n -vertex graph contains at most O ( 1 8 n / 5 ) ≤ O ( 1.78 3 n ) Hamilton cycles, which improves a previous bound by Sharir and Welzl. From the other side we exhibit a family of graphs with 4 8 n / 8 ≥ 1.62 2 n Hamilton cycles per graph. Moreover, we present an algorithm which finds the minimum weight Hamilton cycle of a given 4-regular graph in time 3 n ⋅ poly ( n ) = O ( 1.73 3 n ) , improving on Eppstein’s previous bound. 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 ( 1.78 3 n ) for the number of Hamilton cycles serves also as a time bound for enumeration.