Sequentially Perfect and Uniform One-Factorizations of the Complete Graph

Date

2005

Authors

Dinitz, Jeffrey H.
Dukes, Peter J.
Stinson, Douglas R.

Journal Title

Journal ISSN

Volume Title

Publisher

The Electronic Journal of Combinatorics

Abstract

In this paper, we consider a weakening of the de nitions of uniform and perfect one-factorizations of the complete graph. Basically, we want to order the 2n − 1 one-factors of a one-factorization of the complete graph K2n in such a way that the union of any two (cyclically) consecutive one-factors is always isomorphic to the same two-regular graph. This property is termed sequentially uniform; if this two-regular graph is a Hamiltonian cycle, then the property is termed sequentially perfect. We will discuss several methods for constructing sequentially uniform and sequentially perfect one-factorizations. In particular, we prove for any integer n 1 that there is a sequentially perfect one-factorization of K2n. As well, for any odd integer m > 1, we prove that there is a sequentially uniform one-factorization of K2tm of type (4; 4; : : : ; 4) for all integers t > 2 + dlog2me (where type (4; 4; : : : ; 4) denotes a two-regular graph consisting of disjoint cycles of length four).

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Citation

Dinitz, J. H., Dukes, P., & Stinson, D. R. (2005). Sequentially Perfect and Uniform One-Factorizations of the Complete Graph. The Electronic Journal of Combinatorics, 12. https://doi.org/10.37236/1898