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 if $q\equiv3$ (mod 4) is a prime power with $q\geq11$, then there is a pair of orthogonal totally $C_4$-free one-factorizations of $K_{q+1}$.

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