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. Here’s a link to a WA group for general discussions and meetup reminders. The Zoom details are below:

Meeting ID: 966 5025 7499
Passcode: FTW

See you there!

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