This is a course intended for first year research students in Mathematics, provided for the London Taught Course Centre (LTCC). See the LTCC website for full details of the objectives and activities of the LTCC, and of other available courses.
The following handout summarises some general information about the course – much of the information in this handout is repeated below.
Our aims in this course are twofold. First, to discuss some of the major results of graph theory, and to provide an introduction to the language, methods and terminology of the subject. Second, to emphasise various approaches (algorithmic, probabilistic, etc) that have proved fruitful in modern graph theory: these modes of thinking about the subject have also proved successful in other areas of mathematics, and we hope that students will find the techniques learnt in this course to be useful in other areas of mathematics.
Below is a collection of books, including some that can be accessed online. Any one of these textbooks should give sufficient background reading material.
Reinhard Diestel, Graph Theory (1st, 2nd, 3rd, or 4th edition). Springer (1997, 2000, 2005, 2010).
Although this book is still in print, the author has made sure that a restricted version is available online as well. See diestel-graph-theory.com/. All editions are suitable for this course. References in the notes will refer to the 4th edition (which is the same as the one you can download most parts of).
Noga Alon and Joel H. Spencer, The Probabilistic Method (1st–4th edition). Wiley (1991–2016).
A great, clearly written, and well-motivated book with applications of probability in combinatorics. Many exercises. You can download the 3rd edition from here.
J.A. Bondy and U.S.R. Murty, Graph Theory. Springer (2008).
A thorough and well-written textbook covering most parts of modern
graph theory. In many institutes you will be able to read this book
online.
Long ago, Bondy and Murty wrote one of the classic
textbooks on graph theory: Graph Theory with Applications. North Holland
(1976). This book is out of print (and has been out of print for ages).
But the full text is available online for personal use. You can download
it from here.
Béla Bollobás, Modern Graph Theory. Springer (1998).
This is another classic textbook aimed at students at this level, and is suitable for the course.
Many people attending the course will have taken an introductory course in graph theory or discrete mathematics before, and we assume a certain amount of basic knowledge in graph theory. Specifically, we expect students attending these lectures to be familiar with the following notions: graphs; trees; paths; cycles; vertex degree; connectedness; bipartite graphs; complete graphs; subgraphs.
Those requiring a quick refresher are advised to look at the introductory chapter of any of the books listed above, before the course starts.
Here the lecture notes accompanying the lectures with exercises will be published throughout the course. We encourage all students to attempt the exercises after the corresponsing lectures.
Week 1: Graph Colouring
Notes, Exercises, and Solutions:
Week 2: Graphs on Surfaces; Graph Minors
Notes, Exercises, and Solutions:
Week 3: Algorithms and Complexity
Notes, Exercises, and Solutions:
Week 4: Probabilistic Methods and Random Graphs
Notes, Exercises, and Solutions:
Week 5: Ramsey Theory and Regularity
Notes, Exercises, and Solutions:
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