Perfect 1-factorizations Academic Article uri icon

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abstract

  • AbstractLetGbe a graph with vertex-setV=V(G) and edge-setE=E(G). A 1-factorofG(also calledperfect matching) is a factor ofGof degree 1, that is, a set of pairwise disjoint edges which partitionsV. A 1-factorizationofGis a partition of its edge-setEinto 1-factors. For a graphGto have a 1-factor, |V(G)| must be even, and for a graphGto admit a 1-factorization,Gmust be regular of degreer, 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 ofGis said to beperfectif the union of any two of its distinct 1-factors is a Hamiltonian cycle ofG. 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.

publication date

  • June 26, 2019