Orthogonal cycle systems with cycle length less than 10

TL;DR

This paper constructs pairs of orthogonal cycle systems with cycle lengths less than 10 for most admissible orders, advancing combinatorial design theory.

Contribution

It provides explicit constructions of orthogonal cycle systems for lengths 5 to 9, excluding two specific cases, filling gaps in combinatorial design literature.

Findings

Constructed orthogonal 5-cycle systems for all admissible orders.

Constructed orthogonal 6, 8, and 9-cycle systems for all admissible orders.

Identified exceptions for 7 and 9-cycle systems at specific orders.

Abstract

An $H$-decomposition of $G$ is a partition of the edge-set of $G$ into subsets, where each subset induces a copy of the graph $H$. A $k$-orthogonal $H$-decomposition of a graph $G$ is a set of $k$ $H$-decompositions of $G$, such that any two copies of $H$ in distinct $H$-decompositions intersect in at most one edge. When $G=K_v$ we call the $H$-decomposition an $H$-system of order $v$. In this paper we consider the case $H$ is an $l$-cycle and construct a pair of orthogonal $l$-cycle systems for all admissible orders when $l=5,6,7, 8\ or\ 9$, except $(l,v)=(7,7)$ and $(l,v)=(9,9)$.