On some properties of cyclic tournaments

• Main Author: Remoto, Shirlee Ribaya
• Format: text
• Language: English
• Published: Animo Repository 1994
• Subjects: Paths and cycles (Graph theory); Graph theory; Combinatorial group theory; Matrices; Homomorphisms (Mathematics); Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_masteral/1561; https://animorepository.dlsu.edu.ph/context/etd_masteral/article/8399/viewcontent/TG02248_F_Redacted.pdf
• Institution: De La Salle University

The thesis presents four main theorems on cyclic tournaments. The first deals with the problem of determining the size of any equivalence class of a tournament A in the set C(v) of all cyclic tournaments of order v. The form of an element of the set W(v) of all subgroups of S(v) of odd orders containing C = (123...v) as an automorphism group for some cyclic tournaments is introduced by the second proposition. This is extended to the form of a maximal element of W(v) which is demonstrated by the third theorem using the Polya composition operation. The last theorem discusses a way to determine if an element of W(v) is of the largest order by a certain linear order of odd primes.The main results presented by Noboru Ito in the article On Cyclic Tournaments are amplified. Illustrations are provided to lend plausibility to the theorems. Related theorems and definitions needed in the subsequent arguments of the study but are not stated in Ito's paper are also presented.Some of the primary properties of cyclic tournaments are proved in this study. In addition, a procedure to construct a cyclic tournament such that the automorphism group contains an element of W(v) is demonstrated.The thesis presents four main theorems on cyclic tournaments. The first theorem deals with the problem of determining the size of any equivalence class of a tournament A in the set C(v) of all cyclic tournaments of order v. The form of an element of the set W(v) of all subgroups of S(v) of odd orders containing C = (123...v) as an automorphism group for some cyclic tournaments is introduced by the second proposition. This is extended to the form of a maximal element of W(v) which is demonstrated by the third theorem using the Polya composition operation. The last theorem discusses a way to determine if an element of W(v) is of the largest order by a certain linear order of odd primes.