A perfect 1-factorization (P1F) of a -regular graph is a proper edge coloring using colors (meaning: two edges that meet at a vertex get different colors) so that the union of any two colors forms a Hamiltonian cycle. A graph with a P1F must have an even vertex count. This Demonstration shows P1Fs for over 1000 graphs in Mathematica's graph database, GraphData. When "polygon view" is chosen, each Hamiltonian cycle arising from a pair of colors is shown as a polygon.

Perfect 1-factorizations are a difficult topic in graph theory, since they are not understood even for complete graphs. The outstanding conjecture is that every even complete graph admits a perfect 1-factorization; the first open case is [1].

The P1Fs were found using a variety of techniques. The 3-regular case is simplest since one need only look for a Hamiltonian cycle such that the two matchings from the cycle together with the matching defined from the complement of the cycle form a P1F. One can search for random Hamiltonian cycles until a P1F is found.

For , one first develops a randomized algorithm for edge colorings, making use of random maximum matchings; that is fast thanks to the famous blossom algorithm. Often an edge coloring leads to a Hamiltonian decomposition (HD), which is a partition of all the edges of a graph into Hamiltonian cycles or, when is odd, a set of Hamiltonian cycles plus one perfect matching. One can then search for a P1F by generating many random HDs and so obtaining a set of matchings by breaking each cycle of each HD apart into two matchings. The collection of matchings so obtained is then turned into a graph by connecting two matchings by an edge if they fit together to form a Hamiltonian cycle. This graph can then be searched for a complete subgraph of size ; if such is found then the corresponding matchings form a P1F.

References

[1] M. Li, J. Shu. "C-Codes: Cyclic Lowest-Density MDS Array Codes Constructed Using Starters for RAID 6." (Jul 18, 2012) arxiv.org/abs/1104.2547.

[3] D. Bryant, B. M. Maenhaut, and I. M. Wanless, "A Family of Perfect Factorisations of Complete Bipartite Graphs," Journal of Combinatorial Theory A,98(2), 2002 pp. 328–342. users.monash.edu.au/~iwanless/papers/perfactfamily.pdf.