# Circular Intensely Orthogonal Double Cover Design of Balanced Complete Multipartite Graphs

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## Abstract

**:**

## 1. Introduction

- (i)
**double cover property:**Every edge of ${K}_{n,n}$ is in exactly one page of $\{{G}_{0},{G}_{1},\dots ,{G}_{n-1}\}$ and in exactly one page of $\{{F}_{0},{F}_{1},\dots ,{F}_{n-1}\};$- (ii)
**orthogonality property:**For $i,j\in \{0,1,\dots ,n-1\}$ and $i\ne j,$ $\left|E({G}_{i})\cap E({G}_{j})\right|=\left|E({F}_{i})\cap E({F}_{j})\right|=0;$ and $\left|E({G}_{i})\cap E({F}_{j})\right|=1$ for all $i,j\in \{0,1,\dots ,n-1\}.$

## 2. Basic Definitions and Terminologies

**Definition**

**1.**

- (1)
**double cover property**: every edge of $X$ appears twice in the collection $T.$- (2)
**intensely orthogonality property:**$$\left|E({G}_{{i}_{0}})\cap E({G}_{{j}_{0}})\right|=\left|E({G}_{{i}_{1}})\cap E({G}_{{j}_{1}})\right|=0,i\ne j\mathrm{and}\left|E({G}_{{i}_{0}})\cap E({G}_{{j}_{1}})\right|=\lambda =\left(\begin{array}{l}m\\ 2\end{array}\right),i,j\in {\mathbb{Z}}_{n}.$$

## 3. CIODCDs by Half Starters Matrices

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- (i)
- $\left|E(G)\right|=\lambda n,$
- (ii)
- The lengths of all edges in$G$are mutually different,$i.e.,\{d(e):e\in E(G)\}={\mathbb{Z}}_{n}\times {\mathbb{Z}}_{\lambda}.$

**Lemma**

**1.**

**Proof.**

**Definition**

**5.**

**Theorem**

**1.**

**Proof.**

## 4. CIODCDs by Symmetric Starters Matrices

**Definition**

**6.**

**Remark**

**1.**

**Definition**

**7.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$mG$ | $m$ disjoint copies of $G$ |

${K}_{m}$ | The complete graph on $m$ vertices |

${P}_{k}$ | The path graph on $k$ vertices |

${C}_{m}$ | The cycle on $m$ vertices |

${K}_{m,n}$ | The complete bipartite graph on $m+n$ vertices partitioned into an $m$-stable set and an $n$-stable set. |

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**Figure 2.**CIODCD of ${K}_{3,3,3}$ by $G\cong {K}_{{}_{1,3}}^{0,1}\cup {P}_{4}^{{}^{0,2}}\cup {K}_{{}_{1,3}}^{1,2}.$

**Figure 3.**Symmetric starter of a CIODCD of ${K}_{4,4,4}$ by ${C}_{4}^{0,1}\cup {C}_{4}^{0,2}\cup {C}_{4}^{1,2}$.

**Figure 4.**Symmetric starter of a CIODCD of ${K}_{5,5,5}$ by ${P}_{6}^{0,1}\cup {P}_{6}^{0,2}\cup {P}_{6}^{1,2}.$

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**MDPI and ACS Style**

Higazy, M.; El-Mesady, A.; Mahmoud, E.E.; Alkinani, M.H.
Circular Intensely Orthogonal Double Cover Design of Balanced Complete Multipartite Graphs. *Symmetry* **2020**, *12*, 1743.
https://doi.org/10.3390/sym12101743

**AMA Style**

Higazy M, El-Mesady A, Mahmoud EE, Alkinani MH.
Circular Intensely Orthogonal Double Cover Design of Balanced Complete Multipartite Graphs. *Symmetry*. 2020; 12(10):1743.
https://doi.org/10.3390/sym12101743

**Chicago/Turabian Style**

Higazy, M., A. El-Mesady, Emad E. Mahmoud, and Monagi H. Alkinani.
2020. "Circular Intensely Orthogonal Double Cover Design of Balanced Complete Multipartite Graphs" *Symmetry* 12, no. 10: 1743.
https://doi.org/10.3390/sym12101743