An algorithm for finding factorizations of complete graphs

We show how to find a decomposition of the edge set of the complete graph into regular factors where the degree and edge-connectivity of each factor is given. Let t, n, k1, k2, . . . , kt, l1, l2, . . . , lt be nonnegative integers such that ∑ i ki = n− 1, if n is odd then each ki is even, and for 1 ≤ i ≤ t, li ≤ ki and li = 0 if ki = 1. We shall show how to find a factorization of Kn that contains an li-edge-connected ki-factor, 1 ≤ i ≤ t. In the case where each ki = li = 2, these factorizations are Hamiltonian decompositions, and there is an ancient and well-known method of construction. This method can be extended in a fairly obvious way to k-regular k-connected factorizations of Kkn+1, as shown in [1]; the only difficulty is showing that the k-factors are k-connected. Being k-connected, they are a fortiori k-edge-connected. The method is further extended in [3] to the