This reviewer usually does not like overly inclusive titles such as the title of this book, preferring titles that inform the reader better about the actual content of the book. However, this collection of surveys from the British Combinatorial Conference in 2009 actually deserves its title. The editors did a remarkably good job of selecting papers covering most of combinatorics, even if the various branches are not represented at the same level.
Graph theory (of various kinds) is the subject of more than half of the papers. The longest and deepest of these is a paper by Bollobás and Riordan on Metrics of Sparse Graphs. In this paper, the authors survey random graph models concerning graphs on vertices whose number of edges is negligible compared to the number of edges of the complete graph on the same vertex set. This paper is for specialists, but most of the others in the volume are accessible to a combinatorialist who is not a specialist of the field. A particularly reader-friendly survey is by Bailey and Cameron, on the Combinatorics of Optimal Designs.
Designs and symmetric structures are the second best-represented area. Algebraically-inclined combinatorialists will enjoy Bonasoli’s survey on graph decompositions and symmetry. Finally, Enumerative Combinatorics is present by a paper of Giménez and Noy, which uses analytic and probabilistic tools to count planar graphs and other families of graphs that are similarly defined on other surfaces.
The majority of the papers in this book are suitable for a reading course or a series of seminar presentations by an instructor or advanced graduate student.
Miklós Bóna is Professor of Mathematics at the University of Florida.
1. Graph decompositions and symmetry
2. Combinatorics of optimal designs
R. A. Bailey and Peter J. Cameron
3. Regularity and the spectra of graphs
W. H. Haemers
4. Trades and t-designs
G. B. Khosrovshahi and B. Tayfeh-Rezaie
5. Extremal graph packing problems: Ore-type versus Dirac-type
H. A. Kierstead, A. V. Kostochka, and G. Yu
6. Embedding large subgraphs into dense graphs
Daniela Kühn and Deryk Osthus
7. Counting planar graphs and related families of graphs
Omer Giménez and Marc Noy
8. Metrics for sparse graphs
B. Bollobás and O. Riordan
9. Recent results on chromatic and flow roots of graphs and matroids
G. F. Royle