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

A pair of orthogonal one-factorizations $${\mathcal {F}}$$ and $${\mathcal {G}}$$ of the complete graph $$K_n$$ is totally $$C_4$$ -free, if the union $$F\cup G$$ , for any $$F,G\in {\mathcal {F}}\cup {\mathcal {G}}$$ , does not include a cycle of length four. In this note, we prove that, if $$q\equiv 3$$ (mod 4) is a prime power with $$q\ge 11$$ , then there is a pair of orthogonal totally $$C_4$$ -free one-factorizations of $$K_{q+1}$$ .