PERFECT 1-FACTORIZATIONS

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, jV (G) j must be even, and for a graph G to admit a 1-factorization, G must be regular of degree r, 1 <= r <= vertical bar V vertical bar - 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.