# Tight Bounds on 1-Harmonious Coloring of Certain Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Lower Bound

**Theorem**

**1.**

#### 2.1. Mesh Network

**Theorem**

**2.**

**Proof.**

#### 2.2. Extended Mesh Network

**Theorem**

**3.**

**Proof.**

#### 2.3. Generalized Honeycomb Network

**Theorem**

**4.**

**Proof.**

## 3. Tight Bounds on Some Special Classes of Graphs

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## 4. The 1-harmonious Coloring of the Butterfly Network

**Theorem**

**8.**

**Theorem**

**9.**

**Proof.**

- (i)
- $c\left(\right[01,1\left]\right)=4$ or 5,
- (ii)
- $c\left(\right[10,1\left]\right)=2$ or 3.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) ${h}_{1}\left({P}_{8}\right)=3$, (

**b**) ${h}_{1}\left({C}_{6}\right)=3$, (

**c**) ${h}_{1}\left({C}_{8}\right)=4$, and (

**d**) ${h}_{1}\left({C}_{5}\right)=5$.

**Figure 2.**(

**a**) ${h}_{1}\left({K}_{5}\right)=5$, (

**b**) ${h}_{1}\left({S}_{6}\right)=6$, and (

**c**) ${h}_{1}\left({W}_{7}\right)=7$.

**Figure 6.**(

**a**) Horizontal edge-cuts of $GH(3,2,1,2,1)$ and (

**b**) ${h}_{1}\left(GH(3,2,1,2,1)\right)=4$.

S. No. | $\mathbf{Graph}\phantom{\rule{4pt}{0ex}}\mathit{G}$ | ${\mathit{h}}_{1}\left(\mathit{G}\right)$ |
---|---|---|

1 | Complete graph | $\mid \phantom{\rule{-2.27626pt}{0ex}}V\left(G\right)\phantom{\rule{-2.27626pt}{0ex}}\mid $ |

2 | Wheel | |

3 | Star | |

4 | Tree | $\Delta \left(G\right)+1$ |

5 | Mesh Network | |

6 | Extended Mesh Network | |

7 | Generalized Honeycomb Network |

S. No. | Row i | Color Order |
---|---|---|

1 | i ($mod$ 5) = 1 | $\u23291,2,3,4,5\u232a$ |

2 | i ($mod$ 5) = 2 | $\u23293,4,5,1,2\u232a$ |

3 | i ($mod$ 5) = 3 | $\u23295,1,2,3,4\u232a$ |

4 | i ($mod$ 5) = 4 | $\u23292,3,4,5,1\u232a$ |

5 | i ($mod$ 5) = 0 | $\u23294,5,1,2,3\u232a$ |

S. No. | Row i | Color Order |
---|---|---|

1 | i ($mod$ 3) = 1 | $\u23291,2,3,4,5,6,7,8,9\u232a$ |

2 | i ($mod$ 3) = 2 | $\u23294,5,6,7,8,9,1,2,3\u232a$ |

3 | i ($mod$ 3) = 0 | $\u23297,8,9,1,2,3,4,5,6\u232a$ |

S. No. | $\mathit{l}\left({\mathit{P}}_{{\mathit{v}}_{1}}^{\mathit{\gamma}}\right)$ | Color Order of ${\mathit{P}}^{\mathit{\gamma}}$ |
---|---|---|

1 | 1 | $\u23291,2,3,4\u232a$ |

2 | 2 | $\u23292,3,4,1\u232a$ |

3 | 3 | $\u23293,4,1,2\u232a$ |

4 | 0 | $\u23294,1,2,3\u232a$ |

S. No. | $\mathit{l}\left(\right[00\dots 0],\mathit{i})$ | Color Order of Level i from Top to Bottom | ||
---|---|---|---|---|

$\mathit{i}=\mathbf{0},\mathbf{2}\mathit{n}$ | $\mathbf{1}\le \mathit{i}\le \mathit{n}-\mathbf{1}$ | $\mathit{n}+\mathbf{1}\le \mathit{i}\le \mathbf{2}\mathit{n}-\mathbf{1}$ | ||

1 | 0 | $\u2329{1}_{{2}^{n-1}},{2}_{{2}^{n-1}}\u232a$ | ${\u2329{1}_{{2}^{n-i-1}},{2}_{{2}^{n-i}},{1}_{{2}^{n-i-1}}\u232a}_{{2}^{i-1}}$ | ${\u2329{1}_{{2}^{i-n-1}},{2}_{{2}^{i-n}},{1}_{{2}^{i-n-1}}\u232a}_{{2}^{2n-i-1}}$ |

2 | 1 | $\u2329{3}_{{2}^{n-1}},{4}_{{2}^{n-1}}\u232a$ | ${\u2329{3}_{{2}^{n-i-1}},{4}_{{2}^{n-i}},{3}_{{2}^{n-i-1}}\u232a}_{{2}^{i-1}}$ | ${\u2329{3}_{{2}^{i-n-1}},{4}_{{2}^{i-n}},{3}_{{2}^{i-n-1}}\u232a}_{{2}^{2n-i-1}}$ |

3 | 2 | $\u2329{5}_{{2}^{n-1}},{6}_{{2}^{n-1}}\u232a$ | ${\u2329{5}_{{2}^{n-i-1}},{6}_{{2}^{n-i}},{5}_{{2}^{n-i-1}}\u232a}_{{2}^{i-1}}$ | ${\u2329{5}_{{2}^{i-n-1}},{6}_{{2}^{i-n}},{5}_{{2}^{i-n-1}}\u232a}_{{2}^{2n-i-1}}$ |

S. No. | Level i | Color Order of Level i | |
---|---|---|---|

$\mathbf{1}\le \mathit{i}\le \mathit{n}-\mathbf{1}$ | $\mathit{i}=\mathit{n}$ | ||

1 | i ($mod$ 3) = 0 | ${\u2329{1}_{{2}^{i-1}},{2}_{{2}^{i}},{1}_{{2}^{{2}^{i-1}}}\u232a}_{{2}^{n-i-1}}$ | $\u2329{1}_{{2}^{n-1}},{2}_{{2}^{n-1}}\u232a$ |

2 | i ($mod$ 3) = 1 | ${\u2329{3}_{{2}^{i-1}},{4}_{{2}^{i}},{3}_{{2}^{{2}^{i-1}}}\u232a}_{{2}^{n-i-1}}$ | $\u2329{3}_{{2}^{n-1}},{4}_{{2}^{n-1}}\u232a$ |

3 | i ($mod$ 3) = 2 | ${\u2329{5}_{{2}^{i-1}},{6}_{{2}^{i}},{5}_{{2}^{{2}^{i-1}}}\u232a}_{{2}^{n-i-1}}$ | $\u2329{5}_{{2}^{n-1}},{6}_{{2}^{n-1}}\u232a$ |

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**MDPI and ACS Style**

Liu, J.-B.; Arockiaraj, M.; Nelson, A.
Tight Bounds on 1-Harmonious Coloring of Certain Graphs. *Symmetry* **2019**, *11*, 917.
https://doi.org/10.3390/sym11070917

**AMA Style**

Liu J-B, Arockiaraj M, Nelson A.
Tight Bounds on 1-Harmonious Coloring of Certain Graphs. *Symmetry*. 2019; 11(7):917.
https://doi.org/10.3390/sym11070917

**Chicago/Turabian Style**

Liu, Jia-Bao, Micheal Arockiaraj, and Antony Nelson.
2019. "Tight Bounds on 1-Harmonious Coloring of Certain Graphs" *Symmetry* 11, no. 7: 917.
https://doi.org/10.3390/sym11070917