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

## TL;DR

This paper proves the existence of pairs of orthogonal totally $C_4$-free one-factorizations for complete graphs of order $q+1$, where $q$ is a prime power congruent to 3 mod 4 and at least 11.

## Contribution

It establishes the existence of such factorizations for a new class of complete graphs based on prime powers congruent to 3 mod 4.

## Key findings

- Existence of orthogonal totally $C_4$-free one-factorizations for $K_{q+1}$ when $q
ot o 4$ (mod 4) and $q extgreater 10$.
- Construction methods for these factorizations based on properties of prime powers.
- Extension of previous results on one-factorizations avoiding 4-cycles.

## Abstract

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 $K_{q+1}$.