# On Coloring Catalan Number Distance Graphs and Interference Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Motivational Factor

## 3. Catalan Numbers

## 4. Variations of ${\mathit{C}}_{\mathit{n}},$ and Prime Numbers

## 5. Hankel Transform of Catalan Sequence

## 6. On $\mathit{k}$-Catalan Numbers

## 7. Computation of $\mathit{\chi}(\mathit{G})$ of Catalan Number Distance Set

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Theorem**

**1.**

**Proof.**

**Problem**

**1.**

**Note**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Problem**

**2.**

**Problem**

**3.**

**Note**

**2.**

**Note**

**3.**

## 8. Graph Coloring for Interference Networks

**Theorem**

**4.**

**Discussion.**

**Theorem**

**5.**

## 9. Some Lower and Upper Bounds for $\mathit{\chi}$

**Theorem**

**6.**

**Proof.**

## 10. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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j | ${\mathit{C}}_{2}(\mathit{j})$ | ${\mathit{C}}_{3}(\mathit{j})$ | ${\mathit{C}}_{4}(\mathit{j})$ | ${\mathit{C}}_{5}(\mathit{j})$ | ${\mathit{C}}_{6}(\mathit{j})$ | ${\mathit{C}}_{7}(\mathit{j})$ |
---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 | 1 |

2 | 2 | 3 | 4 | 5 | 6 | 7 |

3 | 5 | 12 | 22 | 35 | 51 | 70 |

4 | 14 | 55 | 140 | 285 | 506 | 819 |

5 | 42 | 273 | 969 | 2530 | 5481 | 10472 |

6 | 132 | 1428 | 7084 | 23751 | 62832 | 141778 |

7 | 429 | 7752 | 53820 | 231880 | 749398 | 1997688 |

8 | 1430 | 43263 | 420732 | 2330445 | 9203634 | 28989675 |

9 | 4862 | 246675 | 3362260 | 23950355 | 115607310 | 430321633 |

10 | 16796 | 1430715 | 27343888 | 250543370 | 1478314266 | 6503352856 |

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**MDPI and ACS Style**

Yegnanarayanan, V.; Yegnanarayanan, G.N.; M. Balas, M.
On Coloring Catalan Number Distance Graphs and Interference Graphs. *Symmetry* **2018**, *10*, 468.
https://doi.org/10.3390/sym10100468

**AMA Style**

Yegnanarayanan V, Yegnanarayanan GN, M. Balas M.
On Coloring Catalan Number Distance Graphs and Interference Graphs. *Symmetry*. 2018; 10(10):468.
https://doi.org/10.3390/sym10100468

**Chicago/Turabian Style**

Yegnanarayanan, Venkataraman, Gayathri Narayana Yegnanarayanan, and Marius M. Balas.
2018. "On Coloring Catalan Number Distance Graphs and Interference Graphs" *Symmetry* 10, no. 10: 468.
https://doi.org/10.3390/sym10100468