Factors and factorizations of graphs. Proof techniques in factor theory
The book covers such central topics of the theory of graph factorization as matchings, regular factors, \(f\)-factors, \((g,f)\)-factors, \([a,b]\)-factorisation. These topics are studied with purely graph theoretic methods that do not involve the theory of matching polytopes. The authors have published extensively in the field of graph factorization, and they demonstrate their deep understanding of the field throughout the book. They show that many of the fundamental theorems of the theory can be derived from a few standard proof techniques. The first two chapters may serve as study material for introductory courses on graph theory whereas the later chapters should be of great value to graduate students and researchers in graph theory. The book is written very carefully and in clear style, and it contains numerous figures illustrating key notions. Plummer and Lovász' book [\textit{L. Lovász} and \textit{M.D. Plummer}, Matching Theory, Ann. Discr. Math. 29, North-Holland Math. Studies 121, Amsterdam (1986; Zbl 0516.05047)] has been the defining work on graph factorization since its publication in 1986. However, several hundred papers on graph factorization have appeared since 1986. Akiyama and Kano's book makes a great contribution to furthering the study of graph factorization by collecting and exhibiting some of the most important concepts and results obtained since the nineteen eighties. The book mentions a number of open problems relating to graph factorization, in particular, the Lovász-Plummer Conjecture; that conjecture was recently solved in the affirmative by \textit{L. Esperet}, \textit{F. Kardoš}, \textit{A.D. King}, \textit{D. Král}, and \textit{S. Norine} [``Exponentially many perfect matchings in cubic graphs,'' Adv. Math. 227, No.\,4, 1646--1664 (2011; Zbl 1223.05229)].
scientific article; zbMATH DE number 5916932