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Alison M. Marr and W. D. Wallis
Publisher: Birkhäuser (2013)
Details: 186 pages, Hardcover
Topics: Combinatorics, Graph Theory
MAA Review[Reviewed by Miklós Bóna, on 02/03/2013]
This is a very basic survey on magic labelings of graphs, which are a special case of the general topic of graph labelings. There is nothing in the book that would not be accessible for an undergraduate student who has taken a few weeks of graph theory.
A graph with v vertices and e edges is called edge magic if it is possible to bijectively label its vertices and edges with the numbers 1 through v+e so that for each edge of the graph, the sum of the labels of the two endpoints and the label of the edge is equal to a constant k. A graph is called vertex magic if a labeling using those same numbers exists so that for each vertex v, the sum of the label of v and of all edges adjacent to v is equal to a constant K. The intriguing question is to decide which graphs are edge magic or vertex magic, or both. The latter are called totally magic graphs.
New to this edition is are many new results for directed graphs, since until the early 2000s, most results concerned undirected graphs. There are a few exercises as well, though usually not more than one per section. More interesting are the research problems that can be found all over the book, and that whose current status is explained in a separate section at the end of the book.
Miklós Bóna is Professor of Mathematics at the University of Florida.