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

TL;DR

This paper constructs specific one-factorizations and pairs of orthogonal one-factorizations of complete graphs with properties related to cycles of length 4, for graphs of size related to primes congruent to 11 mod 24.

Contribution

It proves the existence of a ($1,C_4$) one-factorization and a pair of orthogonal ($2,C_4$) one-factorizations of $K_{q+1}$ for primes $q ot eq 11$ with $q mod 24 = 11$, expanding combinatorial design theory.

Findings

Existence of ($1,C_4$) one-factorization for $K_{q+1}$ when $q ot eq 11$ and $q mod 24=11$.

Existence of a pair of orthogonal ($2,C_4$) one-factorizations for $K_{q+1}$ under the same conditions.

The results apply to complete graphs of size related to specific prime powers.

Abstract

An one-factorization $\mathcal{F}$ of the complete graph $K_n$ is ($l,C_k$), where $l\geq0$ and $k\geq4$ are integers, if the union $F\cup G$, for any $F,G\in\mathcal{F}$, includes exactly $l$ (edge-disjoint) cycles of length $k$ ($lk\leq n$). 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 $q\equiv11$ (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 $K_{q+1}$.