On ($1,C_4$) One-Factorization and Two Orthogonal ($2,C_4$) One-Factorizations of Complete Graphs

Abstract

A 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 $K_{q+1}$.