Pearls in Graph Theory

Pearls in Graph Theory: A Comprehensive Introduction is an undergraduate-level textbook on graph theory, by Gerhard Ringel and Nora Hartsfield. It was published in 1990 by Academic Press, Inc.,[1][2][3] with a revised edition in 1994[4] and a paperback reprint of the revised edition by Dover Books in 2003.[5] The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.[5]

Topics

The "pearls" of the title include theorems, proofs, problems, and examples in graph theory. It has ten chapters; after an introductory chapter on basic definitions, the remaining chapters material on graph coloring; Hamiltonian cycles and Euler tours; extremal graph theory; subgraph counting problems including connections to permutations, derangements, and Cayley's formula; graph labelings; planar graphs, the four color theorem, and the circle packing theorem; near-planar graphs; and graph embeddings on topological surfaces.[4][5] The book also includes several unsolved problems such as the Oberwolfach problem on covering complete graphs by cycles, the characterization of magic graphs, and ringel's "earth-moon" problem on coloring biplanar graphs.[3]

Despite its subtitle promising "a comprehensive introduction" to graph theory, many important topics in graph theory are not covered,[1][4] with the selection of topics reflecting author Ringel's research interests.[1][5] Missing topics include the symmetries of graphs, cliques, connections between graphs and linear algebra including adjacency matrices, algebraic graph theory and spectral graph theory, the connectivity of graphs (or even its Biconnected components), Hall's marriage theorem, line graphs, interval graphs, and the theory of tournaments. There is also only a single chapter of coverage on algorithms and real-world applications of graph theory.[1][4][5] Additionally, the book skips over "difficult or long proofs".[2][5]

Audience and reception

Although the book is written as a lower-level undergraduate textbook, and recommends that students using it have previously taken a course in discrete mathematics, it can be read and understood by students with only a high school background in mathematics. Reviewer L. W. Beineke writes that the variety of levels of the exercises are one of the strengths of the book,[4] and reviewer John S. Maybee writes that they are "extensive" and provide interesting connections to additional topics;[1] however, reviewer J. Sedláček criticizes them as "routine".[2]

Although multiple reviewers complained about the book's spotty or missing coverage of important topics,[1][4][5] reviewer Joan Hutchinson praised its choice of topics as "refreshingly different" and noted that, among many previous texts on graph theory, none had as much depth of coverage of topological graph theory.[3] Other reviewer complaints include a misattributed example,[2] a bad definition of the components of a graph that failed to apply to graphs with one component,[5] and a proof of the five-color theorem that only applies to special planar maps instead of all planar graphs.[3]

Despite these complaints, Beineke writes that, as an undergraduate text, "this book has much to offer".[4] Maybee writes that the book was "a joy to read", provided better depth of coverage on some topics than previous graph theory texts, and would be helpful reading for "many graph theorists".[1] And Hutchinson praises it as providing "a splendid, enticingly elementary yet comprehensive introduction to topological graph theory".[3]

References
  1. "Review of Pearls in Graph Theory (1st ed.)", SIAM Review, 33 (4): 664–665, December 1991, JSTOR 2031030
  2. Sedláček, J., "Review of Pearls in Graph Theory (1st ed.)", zbMATH, Zbl 0703.05001
  3. Hutchinson, Joan P. (November 1991), "Review of Pearls in Graph Theory (revised ed.)", American Mathematical Monthly, 98 (9): 873–875, doi:10.2307/2324291, JSTOR 2324291
  4. Beineke, L. W. (March 1996), "Review of Pearls in Graph Theory (revised ed.)", SIAM Review, 38 (1): 159, JSTOR 2132980; see also Beineke's shorter review in MR1282717
  5. Hunacek, Mark (September 2015), "Review of Pearls in Graph Theory (Dover ed.)", MAA Reviews, Mathematical Association of America
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