ON 4-SEMIREGULAR 1-FACTORIZATIONS OF COMPLETE GRAPHS AND COMPLETE BIPARTITE GRAPHS

A 4-semiregular 1-factorization is a 1-factorization in which every pair of distinct 1-factors forms a union of 4-cycles. Let K be the complete graph K2n or the complete bipartite graph K(n,n). We prove that there is a 4-semiregular 1-factorization of K if and only if n is a power of 2 and n greater-than-or-equal-to 2, and 4-semiregular 1-factorizations of K are isomorphic, and then we determine the symmetry groups. They are known for the case of the complete graph K2n, however, we prove them in a different method.