One-factorizations of regular graphs of order 12 (Q1773192)

The number of non-isomorphic one-factorizations of \(r\)-regular graphs of order 12 is computed for each possible degree \(r,\) with the exception of \(r=7,8,9\), which remains open after this study. In particular a new approach to one-factorizations of \(K_{12}\) that makes use of triple systems is presented. Some properties of the classified one-factorizations (for instance dealing with the structure of their automorphism group) are also tabulated. The algorithms for graphs with small degree are based on viewing one-factorizations as certain error-correcting codes. Two algorithms are used, one that proceeds a one-factor at a time and one that proceeds a vertex at a time. The discussed algorithms do not utilize a classification of regular graphs.