In this fourth edition there are few substantial additions of new material, but many improvements. As with previous new editions, there are countless small and subtle changes to further elucidate a particular argument or concept.

## Book Description

When prompted by reader feedback, for which I am always grateful, I still try to recast details that have been found harder than they should be. These can be very basic; a nice example, this time, is the definition of a minor in Chapter 1.At a more substantial level, there are several new and simpler proofs of classical results, in one case reducing the already shortened earlier proof to half its length (and twice its beauty). These newly added proofs include the marriage theorem, the tree-packing theorem, Tutte’s cycle space and wheel theorem, Fleischner’s theorem on Hamilton cycles, and the threshold theorem for the edge probability guaranteeing a specified type of subgraph. There are also one or two genuinely new theorems. One of these is an ingenious local degree condition for the existence of a

Hamilton cycle, due to Asratian and Khachatrian, that implies a number of classical hamiltonicity theorems.

## Table of Contents

- The Basics
- Matching, covering and packing
- Connectivity
- Planar graphs
- Colouring
- Flows
- Extremal graph theory
- Infinite graphs
- Ramsey theory for graphs
- Hamilton cycles
- Random graphs
- Minors, trees and WQO