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This book surveys matching theory, with an emphasis on connections with other areas of mathematics and on the role matching theory has played, and continues to play, in the development of some of these areas. Besides basic results on the existence of matchings and on the matching structure of graphs, the impact of matching theory is discussed by providing crucial special cases and nontrivial examples on matroid theory, algorithms, and polyhedral combinatorics. The new Appendix outlines how the theory and applications of matching theory have continued to develop since the book was first published in 1986, by launching (among other things) the Markov Chain Monte Carlo method.
This study of matching theory deals with bipartite matching, network flows, and presents fundamental results for the non-bipartite case. It goes on to study elementary bipartite graphs and elementary graphs in general. Further discussed are 2-matchings, general matching problems as linear programs, the Edmonds Matching Algorithm (and other algorithmic approaches), f-factors and vertex packing.