# D-Magic Oriented Graphs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Definitions and Notation

#### 2.1. Oriented Graphs

#### 2.2. Partitions

**Definition**

**2**

#### 2.3. D-Magic Oriented Graphs

**Definition**

**3.**

## 3. Properties of $\mathit{D}$-Magic Oriented Graphs

**Remark**

**1.**

**Lemma**

**1.**

**Proof.**

$f\left(v\right)$ | = | $2k+1$ | |

$f\left(w\right)$ | = | $4k+2$ | |

$f\left({x}_{i}\right)$ | = | $k+i$ | $1\le i\le k$ |

$f\left({x}_{i}\right)$ | = | $4k+2-i$ | $k+1\le i\le 2k$ |

$f\left({y}_{i}\right)$ | = | $k-i+1$ | $1\le i\le k$ |

$f\left({y}_{i}\right)$ | = | $2k+1+i$ | $k+1\le i\le 2k.$ |

**Theorem**

**2.**

**Proof.**

- If $k=1$ and $0\in D$ then $w\left({v}_{2}\right)\ge 2$ and thus ${v}_{2}$ would not have the magic constant as its weight. Thus, $0\notin D$, but then there is no way for $w\left({v}_{1}\right)=1$.
- If $k=2$ and $0\in D$, $w\left({v}_{1}\right)\ne 2$. Thus, $0\notin D$, but then $w\left({v}_{2}\right)\ne 2$ as there are no sums of the remaining vertex labels that would total to two.
- If $k=3$ and $0\in D$, then in order for $w\left({v}_{1}\right)=w\left({v}_{2}\right)=3$ we would need both ${v}_{1}\to {v}_{2}$ and ${v}_{2}\to {v}_{1}$, creating a bidirectional edge. Since $0\notin D$, we would have that ${v}_{3}$ would need to be some distance ${d}_{1}\in D$ away from ${v}_{1}$. However, ${v}_{1}$ would also have to be distance ${d}_{2}\in D$ away from ${v}_{3}$ (as this is the only way for $w\left({v}_{1}\right)=3$). Suppose without loss of generality that ${d}_{2}\le {d}_{1}$. If ${d}_{1}={d}_{2}$, then ${d}_{1}>1$ and we would have a vertex x on the path from ${v}_{1}$ to ${v}_{3}$ that would be distance ${d}_{1}$ away from something larger than three, and thus $w\left(x\right)>3$. If ${d}_{2}<{d}_{1}$, then ${v}_{1}$ would also be distance ${d}_{1}$ away from something other than ${v}_{3}$, giving $w\left({v}_{1}\right)>3$. Thus, in all cases, we arrive at a contradiction.
- A similar argument holds if $k=4$ using ${v}_{1}$ and ${v}_{3}$ (instead of ${v}_{1}$ and ${v}_{2}$).

**OpenQuestion**

**1.**

**OpenQuestion**

**2.**

**Theorem**

**3.**

**Proof.**

**OpenQuestion**

**3.**

**OpenQuestion**

**4.**

## 4. Classes of $\mathit{D}$-Magic Oriented Graphs

#### 4.1. Trees

**Lemma**

**2.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 4.2. Cycles

**Lemma**

**3.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**OpenQuestion**

**5.**

#### 4.3. Multipartite Graphs

**Theorem**

**6.**

**Proof.**

**OpenQuestion**

**6.**

**Theorem 7.**

**Proof.**

**OpenQuestion**

**7.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Marr, A.; Simanjuntak, R. *D*-Magic Oriented Graphs. *Symmetry* **2021**, *13*, 2261.
https://doi.org/10.3390/sym13122261

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Marr A, Simanjuntak R. *D*-Magic Oriented Graphs. *Symmetry*. 2021; 13(12):2261.
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Marr, Alison, and Rinovia Simanjuntak. 2021. "*D*-Magic Oriented Graphs" *Symmetry* 13, no. 12: 2261.
https://doi.org/10.3390/sym13122261