A note on two orthogonal totally $$C_4$$-free one-factorizations of complete graphs

On ($1,C_4$) one-factorization and two orthogonal ($2,C_4$) one-factorization of complete graphs

“…In this section we will prove that the strong starter given by Horton (see Proposition 1.2) doesn't generate two orthogonal (1, C 4 ) one-factorization of the complete graph; however, does generate two orthogonal C 4 -free (see for instance [4,30]) and (2, C 4 ) one-factorizations of complete graphs.…”

“…, F n } of a graph G is (l, C k ), where l ≥ 0 and k ≥ 4 integers, if F i ∪ F j include exactly l cycles of length k (lk ≤ n). In particular, if l = 0, we said that the one factorization is C k -free, see for example [4,18,30].…”

“…Strong starters were first introduced by Mullin and Stanton in [25] in constructing of Room squares. Starters and strong starters have been useful to built many combinatorial designs such as Room cubes [10], Howell designs [3], Kirkman triple systems [17,22], Kirkman squares and cubes [24,26], and factorizations of complete graphs [2,4,9,11,15,30].…”

A note on strong Skolem starters

“…Strong starters were first introduced by Mullin and Stanton in [27] in constructing Room squares. Starters and strong starters have been useful to construct many combinatorial designs such as Room cubes [11], Howell designs [3,19], Kirkman triple systems [19,24], Kirkman squares and cubes [25,28], and factorizations of complete graphs [2,4,9,10,12,15,16,20,31].…”