The Interchange Graphs of Non-isomorphic Tournaments with Minimum Score Vectors Are Exactly Hypercubes

• Main Authors: Yi-Jin Liu, 呂宜錦
• Other Authors: An-Hang Chen
• Format: Others
• Language: zh-TW
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/35478602496317352048

碩士 === 國立臺北商業技術學院 === 資訊與決策科學研究所 === 99 === A tournament of size n, denoted by Tn, represents the players p1,p2,...,pn in a round robin tournament and every two distinct players pi and pj compete exactly one game to decide the winner (and the loser) between them and tie is not permitted. If pi beats pj, we write pi→pj. The score of a player pi in a tournament, denoted si, is the number of players beaten by pi, and the score sequence of Tn is a non-decreasing order list of scores of all players, denote by Sn=(s1,s2,...,sn). Let T(Sn) be the collection of tournaments that realize a given score sequence Sn. A tournament is called strong if there exist directed paths for each of a pair of vertices. A score sequence Sn is said to be strong if there is a strong tournament in T(Sn). In a strong tournament Tn with score sequence Sn=(s1,s2,...,sn), Moon shows that there has exactly C(n,3)-Σ(i=1 to n)C(si,2) (directed) cycles of length 3, for short a 3-cycles. A △-interchange is a transformation which reverses the orientations of the arcs in the 3-cycle of a tournament. An interchange graph is an undirected graph whose vertices are the tournaments in T(Sn) and an edge joins two vertices (tournaments) if they can be transformed to each other by a △-interchange. Chen et al., in 2009, shown that the interchange graphs of tournaments with score sequence Ŝn=(1,1,2,...,n-3,n-2,n-2) are hypercubes with dimension n-2. They studied in the case when the vertices of the tournaments were labeled. If the label removed, some of the tournaments can be regarded as the same. In general, two tournaments are said to be isomorphic if there is a one-to-one correspondence between their vertices and edges such that incidences are preserved. In this thesis, we prove that the interchange graph of non-isomorphic tournaments with the same score sequence Ŝn can be hypercube Qn-4.