A one-factorization of a finite simple graph is a set partition of its edge
set into perfect matchings, also called 1-factors
(Wallis 1997; Gill and Wanless 2020, p. 177). Each graph
vertex is incident to exactly one graph edge in
each factor. Thus a graph with a graph
edge and a one-factorization is regular. For
an 
-regular graph with 
, a one-factorization is equivalent to a proper
coloring of its edges with 
colors.
For example, the complete graph 
on vertices 1, 2, 3, and
4 has the three factors 
, 
, and 
, where 
denotes the graph edge joining
and 
.
The union of any two of these factors is a Hamiltonian
cycle, so this is also a perfect one-factorization.
."
Bull. Austral. Math. Soc. 101, 177-185, 2020.