Tournaments and a Fibonacci Link

Tournaments and a Fibonacci Link poster

PROJECT OF MERIT

Research Authorship:

Michael Long, Dr. Daniela Genova

Faculty Mentor:

Dr. Daniela Genova | College of Arts and Sciences | Department of Mathematics and Statistics

Abstract:

Round robin tournaments are a type of directed graphs with applications to athletic competitions and transportation logistics. The presentation begins with a brief series of informative theorems and properties of directed graphs, which are imperative to our understanding of the properties that make directed graphs (and, subsequently, round robin tournaments) uniquely interesting. We then present a number of results about the properties of tournaments (defined as a complete directed graph), including transitivity–a relatively uncommon property used to determine domination in a round robin tournament–and connectivity, which can most often be seen in determining means of transportation between any two locations. Further, we identify a number of applications of round robin tournaments, such as structural formatting of athletic competitions (like those used in the Olympics to determine playoff scheduling for team sports) and transportation logistics (e.g., the grid patterns used in civil engineering for road design in large cities). From here, an observation about unique Hamiltonian cycles in n-tournaments–first published by R.J. Douglas in the form of an identity and later demonstrated through binary matrices by M.R. Garey–and their relationship to the Fibonacci numbers are also discussed. We find that the uniqueness of Hamiltonian cycles–those that encompass the entirety of the vertices of a tournament–are directly tied to Fibonacci’s enamoring numbers.

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