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

Abstract

An one-factorization $\mathcal{F}$ of the complete graph $K_n$ is ($l,C_k$), where $l\ge 0$ and $k\ge 4$ are integers, if the union $F\cup G$, for any $F,G\in \mathcal{F}$, includes exactly $l$ (edge-disjoint) cycles of length $k$ ($lk\le n$). Moreover, a pair of orthogonal one-factorizations $\mathcal{F}$ and $\mathcal{G}$ of the complete graph $K_n$ is ($l,C_k$) if the union $F\cup G$, for any $F\in \mathcal{F}$ and $G\in \mathcal{G}$, includes exactly $l$ cycles of length $k$. In this paper, we prove the following: if $q\equiv 11$ (mod 24) is an odd prime power, then there is a ($1,C_4$) one-factorization of $K_{q+1}$. Also, there is a pair of orthogonal ($2,C_4$) one-factorization of $K_{q+1}$.

arXiv admin note: text overlap with arXiv:1906.09291