Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science and technology. It is being actively used in fields as varied as biochemistry (genomics), electrical engineering (communication networks and coding theory), computer science (algorithms and computation) and operations research (scheduling). The powerful combinatorial methods found in graph theory have also been used to prove fundamental results in other areas of pure mathematics. This book, besides giving a general outlook of these facts, includes new graph theoretical proofs of Fermat's Little Theorem and the Nielson-Schreier Theorem. New applications to DNA sequencing (the SNP assembly problem) and computer network security (worm propagation) using minimum vertex covers in graphs are discussed. We also show how to apply edge coloring and matching in graphs for scheduling (the timetabling problem) and vertex coloring in graphs for map coloring and the assignment of frequencies in GSM mobile phone networks. Finally, we revisit the classical problem of finding re-entrant knight's tours on a chessboard using Hamiltonian circuits in graphs.

This text offers the most comprehensive and up-to-date presentation available on the fundamental topics in graph theory. It develops a thorough understanding of the structure of graphs, the techniques used to analyze problems in graph theory and the uses of graph theoretical algorithms in mathematics, engineering and computer science. There are many new topics in this book that have not appeared before in print: new proofs of various classical theorems, signed degree sequences, criteria for graphical sequences, eccentric sequences, matching and decomposition of planar graphs into trees. Scores in digraphs appear for the first time and include new results due to Pirzada. The climax of the book is a new proof of the famous four colour theorem due to Dharwadker.