# Edge Even Graceful Labeling of Polar Grid Graphs

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Polar Grid Graph ${P}_{m,n}$

**Theorem**

**1.**

**Proof.**

**Illustration.**

**Remark**

**1.**

- ${f}^{\ast}({v}_{1})\equiv 4n+12;\hspace{0.17em}{f}^{\ast}({v}_{i})\equiv 16i-2,\hspace{0.17em}2\le i\le \frac{n}{2};\hspace{0.17em}{f}^{\ast}({v}_{\frac{n}{2}+i})\equiv 16i+6,\hspace{0.17em}1\le i\le \frac{n}{2}-1;$
- ${f}^{\ast}({v}_{n})\equiv 4;\hspace{0.17em}{f}^{\ast}({v}_{i}^{\prime})\equiv 16i+2,\hspace{0.17em}1\le i\le \frac{n}{2}-1;\hspace{0.17em}{f}^{\ast}({{v}^{\prime}}_{\frac{n}{2}+i})\equiv 2;\hspace{0.17em}{f}^{\ast}({{v}^{\prime}}_{\frac{n}{2}+i})\equiv 16i+10,\hspace{0.17em}1\le i\le \frac{n}{2}-2;$
- ${f}^{\ast}({{v}^{\prime}}_{n-1})\equiv 4n-4;{f}^{\ast}({{v}^{\prime}}_{n})\equiv 4n+8\hspace{0.17em}\mathrm{and}\hspace{0.17em}{f}^{\ast}({v}_{0})\equiv 4n+4$.

**Theorem**

**2.**

**Proof.**

**Illustration.**

**Theorem**

**3.**

**Proof.**

**Illustration.**

**Remark**

**2.**

**Illustration.**

**Theorem**

**4.**

**Proof.**

**Remark**

**3.**

**Illustration.**

**Remark**

**4.**

**Remark**

**5.**

**Illustration.**

## 3. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The edge even graceful labeling of the polar grid graphs ${P}_{14,6},\hspace{0.17em}{P}_{16,6},\hspace{0.17em}{P}_{18,6},\hspace{0.17em}{P}_{24,6}$ and ${P}_{26\hspace{0.17em},\hspace{0.17em}6}$.

**Figure 6.**The edge even graceful labeling of the polar grid graphs ${P}_{13,6},\hspace{0.17em}{P}_{15,6},\hspace{0.17em}{P}_{17,6}$ and ${P}_{25,6}$.

**Figure 8.**The edge even graceful labeling of the polar grid graphs ${P}_{10,5},\hspace{0.17em}{P}_{12,5},\hspace{0.17em}{P}_{14,5},\hspace{0.17em}{P}_{16,5},\hspace{0.17em}{P}_{18,5}$ and ${P}_{20,5}$.

**Figure 11.**The polar grid graphs ${P}_{3,5},{P}_{13,5},\hspace{0.17em}{P}_{11,5},\hspace{0.17em}{P}_{7,9},\hspace{0.17em}{P}_{15,5}$ and ${P}_{7,5}.$

**Figure 13.**The polar grid graphs ${P}_{3,7},\hspace{0.17em}{P}_{13,7},\hspace{0.17em}{P}_{19,7},{P}_{11,7}$ ${P}_{15,7}$.

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Daoud, S.N.
Edge Even Graceful Labeling of Polar Grid Graphs. *Symmetry* **2019**, *11*, 38.
https://doi.org/10.3390/sym11010038

**AMA Style**

Daoud SN.
Edge Even Graceful Labeling of Polar Grid Graphs. *Symmetry*. 2019; 11(1):38.
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**Chicago/Turabian Style**

Daoud, Salama Nagy.
2019. "Edge Even Graceful Labeling of Polar Grid Graphs" *Symmetry* 11, no. 1: 38.
https://doi.org/10.3390/sym11010038