Decomposition of K2n into n−1 Hamiltonian cycles and a perfect matching Mi

Theorem on 2-factorization of complete graph K
2n
of even order is proved. Foundamental concepts of 1-factorization and 2-factorization of complete graphs K
2n
are described. Construct the edge matrix of the complete graph K
2n
, edge coloring has been followed, Thus the rectangular matrix has been composed by 1-factor of 2n−1. Determine the collocation programming which is let 2n−1 1-factor combined with n−1 2-factor and 1 1-factor and generated cycles for each 2-factor, In accordance with the different Programming, The process which let E(G) divide into n−1 frontier set of Hamiltonian cycles and 1 1-factor. The entire procedure of 2-factorization of complete graphs K′
10
, K
12
and K
14
is presented.