One-factorizations of regular graphs of order 12

Abstract

Algorithms for classifying one-factorizations of regular graphs are studied. The smallest open case is currently graphs of order 12; one-factorizations of r-regular graphs of order 12 are here classified for r ≤ 6 and r = 10, 11. Two different approaches are used for regular graphs of small degree; these proceed one-factor by one-factor and vertex by vertex, respectively. For degree r = 11, we have one-factorizations of K12. These have earlier been classified, but a new approach is presented which views these as certain triple systems on 4n - 1 points and utilizes an approach developed for classifying Steiner triple systems. Some properties of the classified one-factorizations are also tabulated.