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, K₁₂ and K₁₄ is presented.