On 4-semiregular 1-factorizations of complete graphs and complete bipartite graphs (Q1323492)

A 4-semiregular 1-factorization is a 1-factorization in which every pair of distinct 1-factors forms a union of 4-cycles. Let \(G\) be the complete graph \(K_{2n}\) or the complete bipartite graph \(K_{n,n}\). The authors prove that there is a 4-semiregular 1-factorization of \(G\) iff \(n\) is a power of 2 and \(n\geq 2\), and 4-semiregular 1-factorizations of \(G\) are isomorphic. They determine the symmetry groups as well.