# 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 $r\leq 6$ 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 $K_{12}$. 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.