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.

‘This book is a beautiful tour through graph theory. The book starts gently and carefully guides readers through the essential topics in the area, providing intuition and motivation at each step, before setting the reader up for more advanced topics. The pacing is ideal for a first course in graph theory and additionally contains many excellent exercises.’

Julian Sahasrabudhe - University of Cambridge