Affiliation:
1. Department of Mathematics and Statistics , McMaster University Hamilton , Ontario Canada
Abstract
Abstract
Let G be a graph with vertex-set V = V(G) and edge-set E = E(G). A 1-factor of G (also called perfect matching) is a factor of G of degree 1, that is, a set of pairwise disjoint edges which partitions V. A 1-factorization of G is a partition of its edge-set E into 1-factors. For a graph G to have a 1-factor, |V(G)| must be even, and for a graph G to admit a 1-factorization, G must be regular of degree r, 1 ≤ r ≤ |V| − 1.
One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs.
A 1-factorization of G is said to be perfect if the union of any two of its distinct 1-factors is a Hamiltonian cycle of G. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs.
It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.
Reference97 articles.
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2. Anderson, B. A.: Finite topologies and Hamiltonian paths, J. Combin. Theory 14 (1973), 87–93.
3. Anderson, B. A.: A class of starter-induced 1-factorizations. In: Graphs and Combinatorics, Lect. Notes in Math. 406, Springer-Verlag, 1974, pp. 180–185.
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