Sequentially perfect and uniform one-factorizations of the complete graph (Q1773191)

The notion of a uniform 1-factorization of the complete graph is relaxed here as follows: A 1-factorization of \(K_{2n}\) is called sequentially uniform if its 1-factors can be ordered as \(F_0,F_1,\dots\), \(F_{2n-2}\) so that for each \(i= 0,\dots, 2n-2\), the graph \(F_i\cup F_{i+1}\) (subscripts taken modulo \(2n- 1\)) is isomorphic to the same 2-regular graph \(G\). If \(G\) is the Hamiltonian cycle, the 1-factorization is termed sequentially perfect. The observation is made that the well-known series GK\((2n)\) provides an example of a sequentially perfect 1-factorization if its 1-factors are taken in its ``natural'' order; this is being contrasted with the notoriously difficult problem of determining the orders for which a perfect 1-factorization exists. Examples are given for sequential 1-factorizations of \(K_{2n}\) of all possible types for orders \(2n\leq 24\). Several deeper results are obtained in Sections 4 and 5 dealing with starters in non-cyclic groups, and with a product construction for sequentially uniform 1-factorizations of type \((4,4,\dots, 4)\), respectively. The statement in Section 2 that the 1-factorization GK\((2n)\) is always uniform is obviously incorrect.