Perfect 1-Factorizations of Graphs
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 .
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.
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