Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
On the number of crossing-free matchings, (cycles, and partitions)
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
The Travelling Salesman Problem in Bounded Degree Graphs
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
An improved exact algorithm for cubic graph TSP
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
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We show that each 4-regular n-vertex graph contains at most O(18^n^/^5)@?O(1.783^n) Hamilton cycles, which improves a previous bound by Sharir and Welzl. From the other side we exhibit a family of graphs with 48^n^/^8=1.622^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(1.733^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(1.783^n) for the number of Hamilton cycles serves also as a time bound for enumeration.