Basic Graph Theory

A gentle introduction to the basics of graph theory, with a focus on beautiful proofs and probabilistic methods.

This text provides a gentle but thorough introduction to the areas of extremal graph theory, Ramsey theory and random graphs, while also covering the fundamental results of classical graph theory. It is aimed at undergraduates and Masters students who want to learn the basics of this vibrant area of mathematics.Over the past few decades, graph theory has developed into one of the central areas of modern mathematics, with close (and growing) connections to areas of pure mathematics such as number theory, probability theory, algebra and geometry, as well as to applied areas such as the theory of networks, machine learning, statistical physics, and biology. It is a young and vibrant area, with several major breakthroughs having occurred in just the past few years. This book offers the reader a gentle introduction to the fundamental concepts and techniques of graph theory, covering classical topics such as matchings, colourings and connectivity, alongside the modern and vibrant areas of extremal graph theory, Ramsey theory, and random graphs. The focus throughout is on beautiful questions, ideas and proofs, and on illustrating simple but powerful techniques, such as the probabilistic method, that should be part of every young mathematician's toolkit.