On (1,C4) One-Factorization and Two Orthogonal (2,C4) One-Factorizations of Complete Graphs

A one -factorization.F of the complete graph IC is (1, Ck), where 1> 0 and k> 4 are integers, if the union F U G, for any F, G E.F, includes exactly 1 (edge -disjoint) cycles of length k (lk < n). Moreover, a pair of orthogonal one-factorizations.F and g of the complete graph IC is (1, Ck) if the union F U G, for any F E.F and G E g, includes exactly 1 cycles of length k. In this paper, we prove the following: if q 11 (mod 24) is an odd prime power, then there is a (1, C4) one -factorization of Kg+1. Also, there is a pair of orthogonal (2, C4) one -factorization of k(q+1).