BOOK REVIEWS 253

GRAPH THEORY

By FRANK HARARY: pp. ix, 274. £5.175. (Addison-Wesley Publishing Co. Inc., Reading, Mass., 1969). This is a very readable account of the present status of . The first chapter is a short historical survey showing the origins of the subject in questions of Downloaded from https://academic.oup.com/blms/article/3/2/253/443462 by guest on 28 September 2021 recreational concerning unicursal paths and the colouring of maps, in the work of Kirchoff on electrical networks, and in Cayley's work on the enumeration of the structural formulae of the saturated hydrocarbons. It mentions also a number of recent development in physics and mathematics in which graph-theoretical notions are important. The second chapter is concerned with definitions and the study of connectedness. It also introduces the reader to Ramsey's problem, Turin's Theorem on triangles, Ulam's Conjecture and the theory of intersection graphs. After chapters on blocks and trees there follows one on connectivity. This is notable for a careful exposition of Menger's Theorem, with its many variations. The sixth chapter introduces the theory of " partitions ". Here the problem is that of constructing a graph with a prescribed number of vertices of each valency. (The graphs of this book have no loops or multiple joins). A necessary and sufficient condition for the existence of such a graph is given, as is an algorithm for constructing such a graph if one exists. The seventh chapter deals with Euler and Hamilton paths. A proof is given of P6sa's Theorem, and other important results are mentioned. The author, quite properly, does not give proofs of all his propositions, for many of them are well-known with proofs readily accessible elsewhere. But he does prove many important theorems of current interest to combinatorialists. The statement on page 68 that" the smallest known nonhamiltonian triply connected planar graph " has 38 points presumably omits the word " cubic ". As it stands it is contradicted by the example of an octahedron with each face subdivided into three triangles. The next chapter deals with " line graphs " or " derived graphs ". It includes a proof of the recently discovered theorem that a given graph can be interpreted as the of a second graph if and only if it does not contain any one of nine " for- bidden " subgraphs. An H-factor of a graph G is a subgraph F in which each vertex of G occurs with valency n. Chapter 9 notes some results on the decomposition of special graphs into 1-factors, and it gives a proof of a necessary and sufficient condition for the existence of a 1 -factor in a given graph. Petersen's graph and theorem are also noted. Perhaps it should have been pointed out that Petersen's Theorem is a simple consequence of the condition just mentioned. A result of the reviewer is wrongly stated in Theorem 9.5. The words " do not" should be inserted in " there exist". The theory of 1-factors leads on to those of arboricity and the coverings of graphs. Then comes the eleventh chapter, dealing with planarity. This includes a proof of 254 BOOK REVIEWS

Kuratowski's Theorem. It includes also a discussion of genus, thickness, coarseness and crossing number. It leads on to a discussion of colouring problems and the Heawood Conjecture, recently proved by Ringel and Youngs. The thirteenth chapter discusses the various matrices that are associated with graphs. It notes the existence of matroids and describes the structure of the " whirl ", a simple non-graphic matroid. There follows a chapter on symmetry. This includes an account of Frucht's Theorem that a graph can be constructed with an automorphism Downloaded from https://academic.oup.com/blms/article/3/2/253/443462 by guest on 28 September 2021 group isomorphic to any given finite abstract group. The next chapter expounds P61ya's theory of enumeration, and the last chapter deals with diagraphs and tournaments. In an appendix the author has provided diagrams of all graphs with p " points " and q "lines ". (p ^ 6, q < 15). Similar pictorial tables are given for digraphs O < 4, q ^ 12), and for trees (p ^ 10). This work should be of great value as a text-book in courses on graph theory. It can indeed be recommended to anyone who wishes to know what that subject is about. W. T. TUTTE.

DIFFERENTIAL GEOMETRY

By J. J. STOKER: pp. xxi, 404; 140s. (-Interscience, , 1969). This is an introductory book intended for advanced undergraduate mathematics students and non-specialists who require some understanding of differential geometry. It presupposes only a slight knowledge of analysis, the necessary back ground of tensor algebra and differential equations being provided in appendices. The book deals essentially with curves and surfaces in E3, although there is a reasonable coverage of intrinsic geometry such as geodesies and singularities of vector fields. There is also a short section on relativity. The principal tool is vector calculus, with tensor calculus and differential forms being introduced only as required in later chapters. An outstanding feature of the book is the large number of results on global geometry. These include the Jordan curve theorem for smooth plane curves (seen as an application of the winding number of an arc), the four vertex theorem, the Hopf-Rinow theorem on completeness, convex domains, the Gauss-Bonnet theorem, Poincare's theorem on the sum of the indices of a vector field on a closed surface, Synge's theorem on orientable surfaces with Gaussian curvature K ^ \/k2, covering surfaces, Hilbert's theorem on surfaces of constant negative curvature in £3, and integral formulas applied to closed surfaces. Proofs are presented with considerable attention to detail so as to assist the reader through difficult sections. The author has produced a text on basic differential geometry which, by prudent choice and ordering of material avoids tedious concern with abstract structure and instead focuses attention on theorems of fundamental importance with immediate appeal to students. A. J. LEDGER