Topics on Tournaments | Apr-May 2025

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Topics on Tournaments

John W. Moon

21 Apr 2025 to 18 May 2025
About the Book

Tournaments are complete directed graphs, and derive their name from how they naturally model outcomes of round-robin tournaments.

The study of tournaments is a rich area of graph theory, with applications to various fields such as sports, voting, and ranking. This will be an exploration of the combinatorial aspects of tournaments.

You can find this book on Project Gutenberg.

Prerequisites

The equivalent of a first course in discrete math at an undergraduate level would be useful background to have. Some familiarity with graphs in general is desirable but not required.

Timings and Venue

We meet on Zoom at 9PM IST for about 40-60 minutes every day according to the schedule below. A link to a Whatsapp group will be posted here soon. If you want a heads up when these details are available, drop your email below.

Date Lecture Slides Notes Video
21 Apr, 2025 Introduction
Basic definitions and fundamental concepts of tournaments
22 Apr, 2025 Irreducible Tournaments
Exploring irreducible tournaments and their properties
23 Apr, 2025 Strong Tournaments
Understanding strong tournaments and their characteristics
24 Apr, 2025 Cycles in a Tournament
Analysis of cycles and their significance in tournaments
25 Apr, 2025 Strong Subtournaments of a Tournament
Examining strong subtournaments and their relationships
26 Apr, 2025 The Distribution of 3-cycles in a Tournament
Statistical analysis of 3-cycles in tournament structures
27 Apr, 2025 Transitive Tournaments
Exploring transitive tournaments and their unique properties
28 Apr, 2025 Sets of Consistent Arcs in a Tournament
Understanding consistent arc sets and their implications
29 Apr, 2025 The Parity of the Number of Spanning Paths of a Tournament
Examining parity properties of spanning paths
30 Apr, 2025 The Maximum Number of Spanning Paths of a Tournament
Analysis of maximum spanning paths in tournament structures
01 May, 2025 An Extremal Problem
Exploring extremal problems in tournament theory
02 May, 2025 The Diameter of a Tournament
Understanding tournament diameter and its implications
03 May, 2025 The Powers of Tournament Matrices
Examining matrix representations of tournaments and their powers
04 May, 2025 Scheduling a Tournament
Practical applications in tournament scheduling
05 May, 2025 Ranking the Participants in a Tournament
Methods and algorithms for ranking tournament participants
06 May, 2025 The Minimum Number of Comparisons Necessary to Determine a Transitive Tournament
Optimization problems in transitive tournaments
07 May, 2025 Universal Tournaments
Exploring the concept and properties of universal tournaments
08 May, 2025 Expressing Oriented Graphs as the Union of Bilevel Graphs
Representation theory for oriented graphs in tournaments
09 May, 2025 Oriented Graphs Induced by Voting Patterns
Applications of tournament theory to voting systems
10 May, 2025 Oriented Graphs Induced by Team Comparisons
Applications of tournament theory to sports and competitions
11 May, 2025 Criteria for a Score Vector
Understanding the mathematical criteria for valid score vectors
12 May, 2025 Score Vectors of Generalizations of Tournaments
Extending score vector concepts to generalized tournaments
13 May, 2025 The Number of Score Vectors
Combinatorial analysis of possible score vectors
14 May, 2025 The Largest Score in a Tournament
Extremal analysis of maximum scores in tournaments
15 May, 2025 A Reversal Theorem
Exploring reversal properties in tournament theory
16 May, 2025 Tournaments with a Given Automorphism Group
Algebraic analysis of tournament symmetries
17 May, 2025 The Group of the Composition of Two Tournaments
Understanding compositional properties of tournament groups
18 May, 2025 The Maximum Order of the Group of a Tournament
Exploring maximum order properties in tournament groups
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