tailieunhanh - Báo cáo toán học: "A Note on the Number of Hamiltonian Paths in Strong Tournaments"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Note on the Number of Hamiltonian Paths in Strong Tournaments. | A Note on the Number of Hamiltonian Paths in Strong Tournaments Arthur H. Busch Department of Mathematics Lehigh University Bethlehem PA 18105 ahb205@ Submitted Sep 20 2005 Accepted Jan 18 2006 Published Feb 1 2006 Mathematics Subject Classifications 05C20 05C38 Abstract We prove that the minimum number of distinct hamiltonian paths in a strong tournament of order n is 5 3 . A known construction shows this number is best possible when n 1 mod 3 and gives similar minimal values for n congruent to 0 and 2 modulo 3. A tournament T V A is an oriented complete graph. Let hp T be the number of distinct hamiltonian paths in T . directed paths that include every vertex of V . It is well known that hp T 1 if and only if T is transitive and Rédei 3 showed that hp T is always odd. More generally if T is reducible . not strongly connected then there exists a set A c V such that every vertex of A dominates every vertex of V A. If we denote the subtournament induced on a set S as T S then it is easy to see that hp T hp T A hp T V A . Clearly this process can be repeated to obtain hp T hp T Al hp T A2 hp T At where T Al . T At are the strong components of T. As a result we generally consider hp T for strong tournaments T. In particular we wish to find the minimal value of hp T as T ranges over all strong tournaments of order n. Moon 1 bounded this value above and below with the following result. Theorem Moon 1 . Let hp n be the minimum number of distinct hamiltonian paths in a strong tournament of order n 3. Then 8 3 3n-3 K 3n-1 for n 0 mod 3 an-1 hp n 3n-1 for n 1 mod 3 9 3n-5 3n-1 for n 2 mod 3 where a p6 and 3 p5 . THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N3 1 This lower bound was used by Thomassen 2 to establish a lower bound for the number of hamiltonian cycles in 2-connected tournaments. Theorem Thomassen 2 . Every 2-connected tournament of order n has at least a 32-1 distinct hamiltonian cycles. We shall prove that the upper .