In graph theory, a 1-factor of a graph is a collection of disjoint edges (a matching). A 1-factorization of a graph G is a collection of 1-factors such that every edge of G is in exactly one of these 1-factors. A graph G is said to be 1-factorable if it admits a 1-factorization.
Example
Draw seven vertices distributed evenly around a circle, and one in the middle, for a total of eight vertices. Join the middle point to any single point on the circle; call this line L. Join points two points on the circle together if and only if they can be joined together with a line orthogonal to L. Since the points were arranged evenly, this will produce a matching (in fact a perfect matching) of these eight vertices.
Now rotate the lines one vertex to the right: Start over again with the eight vertices as described and join the center point to the point in the circle directly clockwise from the first one chosen. Join the other points on the circle in a similar matter as before. This is again another perfect matching of these eight points.
(Image: The process described above is shown in the image, with different colors for each of the different 1-Factors. Note the black lines simply show the original circle and should not be interpreted as edges)
Each of these perfect matchings can also be looked at as a 1-factor of the complete graph on eight vertices, K8. Continuing the process above, you will form a 1-factorization of K8. This is a proof that there exists a 1-factorization of for all n.
