A Course in Combinatorics and Graphs

This compact textbook consists of lecture notes given as a fourth-year undergraduate course of the mathematics degree at the Universitat Politècnica de Catalunya, including topics in enumerative combinatorics, finite geometry, and graph theory. This text covers a single-semester course and is aimed at advanced undergraduates and masters-level students. Each chapter is intended to be covered in 6-8 hours of classes, which includes time to solve the exercises. The text is also ideally suited for independent study. Some hints are given to help solve the exercises and if the exercise has a numerical solution, then this is given. The material covered allows the reader with a rudimentary knowledge of discrete mathematics to acquire an advanced level on all aspects of combinatorics, from enumeration, through finite geometries to graph theory.

The intended audience of this book assumes a mathematical background of third-year students in mathematics, allowing for a swifter useof mathematical tools in analysis, algebra, and other topics, as these tools are routinely incorporated in contemporary combinatorics. Some chapters take on more modern approaches such as Chapters 1, 2, and 9. The authors have also taken particular care in looking for clear concise proofs of well-known results matching the mathematical maturity of the intended audience.

Preface.- Chapter 1 Symbolic Enumeration.- Chapter 2 Labelled enumeration.- Chapter 3 Enumeration with symmetries.- Chapter 4 Finite Geometries and Latin Squares.- Chapter 5 Matchings.- Chapter 6 Connectivity.- Chapter 7 Planarity.- Chapter 8 Graph Colouring.- Chapter 9 Extremal Graph Theory.- Chapter10 Hints and solutions to selected exercises.- Bibliography.

Simeon Ball is senior lecturer at the Department of Mathematics of the Universitat Politècnica de Catalunya in Barcelona. He has been invited speaker at many international conferences, as well as serving on the scientific and organising committee for many conferences. His research interests include classical and quantum error-correcting codes, incidence problems in real and finite geometries, graphs and semifields, and is particularly focused on applying geometrical and algebraic methods to these combinatorial objects. He has published more than 70 research papers and two books and serves on the editorial board of the Journal of Geometry and the Journal of Combinatorial Theory Series A, having previously served on the editorial board of Designs, Codes and Cryptography and Finite Fields and Their Applications.

Oriol Serra is full professor at the Department of Mathematics of the Universitat Politècnica de Catalunya in Barcelona. His research interestsare in combinatorics, graph theory and combinatorial number theory. He has been particularly interested in isoperimetric problems, Ramsey Theory, extremal and probabilistic combinatorics and additive combinatorics. He has published more than   100 research and divulgation papers, has been invited speaker in key conferences and has been involved in the organization of several of them. He has been involved in teaching Combinatorics and Graph Theory at undergraduate and master levels, particularly in a successful course on the topic jointly with Prof. Simeon Ball. He has served as chair of the department of mathematics, vicedean of research of the School of Mathematics at UPC and member of the Executive Committee of the Barcelona Graduate School of Mathematics among other duties.