Kfactors
In graph theory, a k-factor of a graph G is a spanning subgraph in which every vertex has degree exactly k. If G has n vertices, a k-factor can exist only when nk is even, since the sum of the degrees in the subgraph must be twice the number of edges. A necessary condition is also that the original graph has minimum degree at least k, since no vertex can gain degree beyond what its incident edges allow.
Special cases include a 1-factor, which is a perfect matching, and a 2-factor, which is a spanning
Existence criteria for general k-factors are captured by Tutte’s f-factor theorem, which provides necessary and sufficient
Applications of k-factors include network design, scheduling, and the study of graph decompositions. See also perfect