# Chromatic Number of Fuzzy Graphs: Operations, Fuzzy Graph Coloring, and Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. K-Coloring of Crisp Graph

#### 2.2. Basic Concepts in Fuzzy Sets

**Definition 1**

**Definition 2**

**Definition 3**

**Definition 4**

**Definition 5**

**Definition 6**

**Definition 7**

#### 2.3. Basic Concepts of Fuzzy Graph Coloring

**Definition 8**

**Definition 9**

**Definition 10**

**Definition 11.**

**Definition 12**

#### 2.4. Operations on Fuzzy Graphs

**Definition 13.**

**Definition 14.**

**Example 1.**

**Definition 15.**

**Definition 16.**

**Definition 17**

**Remark 1.**

**Definition 18.**

**Definition 19.**

**Definition 20.**

**Remark 2.**

**Definition 21.**

**Example 2.**

**Definition 22.**

**Definition 23.**

**Definition 24.**

**Definition 25.**

**Example 3.**

## 3. Chromatic Number of Fuzzy Graphs

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Theorem 1.**

**Proof.**

**Lemma 3.**

**Proof.**

**Corollary 1.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

- (i)
- There do not form new cycles in fuzzy graphs $\tilde{G}(V,\tilde{E})$. It is obvious that $\chi \left(\tilde{G}\right)=max\{\chi \left({\tilde{G}}_{1}\right),\chi \left({\tilde{G}}_{2}\right)\}$.
- (ii)
- There are new cycles formed in the fuzzy graph $\tilde{G}(V,\tilde{E})$. In order to satisfy $f\left(u\right)\ne f\left(v\right)$, new colors may be required in $\tilde{G}$. It is obvious that $max\{\chi \left({\tilde{G}}_{1}\right),\chi \left({\tilde{G}}_{2}\right)\}\le \chi \left(\tilde{G}\right)\le \chi \left({\tilde{G}}_{1}\right)+\chi \left({\tilde{G}}_{2}\right)$.

**Corollary 2.**

**Theorem 4.**

**Proof.**

**Theorem 5.**

**Proof.**

**Theorem 6.**

**Proof.**

**Theorem 7.**

**Proof.**

**Remark 3.**

**Theorem 8.**

**Proof.**

**Theorem 9.**

**Proof.**

**Theorem 10.**

**Proof.**

**Theorem 11.**

**Proof.**

## 4. Application of Chromatic Number of Fuzzy Graphs

#### 4.1. The Examination Problem

**Example 4.**

#### 4.2. The Traffic Light System Problem

- (i)
- If traffic movements $AB$ and $CD$ are incompatible, then there is an edge $(AB,CD)$. In addition, we choose a maximum value of membership degree of $AB$ and $CD$ to determine the membership degree of the fuzzy edge $(AB,CD)$.
- (ii)
- If the traffic movements $AB$ and $CD$ are compatible, which means they can move at the same time, then there is no $(AB,CD)$ edge. This indicates that the fuzzy edge’s membership degree is 0.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) ${\tilde{G}}_{1}\sqcap {\tilde{G}}_{2}$, (

**b**) ${\tilde{G}}_{1}\u25ca{\tilde{G}}_{2}$, (

**c**) ${\tilde{G}}_{1}\otimes {\tilde{G}}_{2}$, and (

**d**) ${\tilde{G}}_{1}\times {\tilde{G}}_{2}$.

**Figure 5.**A traffic light system on two intersections (A–F represent one direction of the intersection respectively).

**Figure 7.**Fuzzy graph models for traffic light system in Figure 5.

**Figure 8.**The model of union of fuzzy graphs in Figure 7.

Slots | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Scheduled exams | H,F,D,B | A | E,G | C |

$\mathit{AB}$ | $\mathit{AD}$ | $\mathit{BA}$ | $\mathit{BC}$ | $\mathit{CD}$ | $\mathit{CA}$ | $\mathit{DC}$ | $\mathit{DB}$ | $\mathit{ED}$ | $\mathit{DF}$ | $\mathit{EF}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

n | 536 | 350 | 476 | 425 | 582 | 396 | 678 | 313 | 275 | 317 | 465 |

Degree (l) | 0.10 | 0.33 | 0.08 | ||||||||

Degree (m) | 0.43 | 0.33 | 0.83 | 0.83 | 0.12 | 0.64 | 0.09 | 0.11 | 0.9 | ||

Degree (h) | 0.07 | 0.47 |

Movements | $\mathit{AB}$ | $\mathit{AD}$ | $\mathit{BA}$ | $\mathit{BC}$ | $\mathit{CD}$ | $\mathit{CA}$ | $\mathit{DC}$ | $\mathit{DB}$ | $\mathit{ED}$ | $\mathit{DF}$ | $\mathit{EF}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

$AB$ | - | - | - | 0.83 | 0.43 | 0.64 | 0.47 | - | - | - | - |

$AD$ | - | - | 0.83 | - | - | 0.64 | 0.47 | 0.33 | - | - | 0.9 |

$BA$ | - | 0.83 | - | - | 0.83 | - | 0.83 | 0.83 | - | - | - |

$BC$ | 0.83 | - | - | - | 0.83 | 0.83 | - | 0.83 | - | - | - |

$CD$ | 0.43 | - | 0.83 | 0.83 | - | - | - | 0.12 | - | 0.12 | 0.9 |

$CA$ | 0.64 | 0.64 | - | 0.83 | - | - | 0.64 | 0.64 | - | - | - |

$DC$ | 0.47 | 0.47 | 0.83 | - | - | 0.64 | - | - | 0.47 | - | 0.9 |

$DB$ | - | 0.33 | 0.83 | 0.83 | 0.12 | 0.64 | - | - | - | - | 0.9 |

$ED$ | - | - | - | - | - | - | 0.47 | - | - | 0.33 | - |

$DF$ | - | - | - | - | 0.12 | - | - | - | 0.33 | - | - |

$EF$ | - | 0.9 | - | - | 0.9 | - | 0.9 | 0.9 | - | - | - |

${\mathit{\alpha}}_{\mathit{i}}$ | k | Arrangements (Partitions of Vertex Set V) |
---|---|---|

0.12 | 4 | $\{AD,CD,ED\},\{BA,CA,DF,EF\},\{BC,DC,\},\{DB,AB\}$ |

0.33 | 4 | $\{AD,CD,ED\},\{BA,CA,DF,EF\},\{BC,DC,\},\{DB,AB\}$ |

0.43 | 4 | $\{AD,CD,ED\},\{BA,CA,DF,EF\},\{BC,DC,\},\{DB,AB\}$ |

0.47 | 3 | $\{AD,DB,AB,EF,ED\},\{BA,BC,DC\},\{DF,CD,CA\}$ |

0.64 | 3 | $\{AD,DB,AB,CD,DC\},\{BC,BA,EF\},\{CA,DF,ED\}$ |

0.83 | 2 | $\{AD,CD,DC,DB,AB\},\{BA,BC,DF,ED,EF\}$ |

0.9 | 2 | $\{AD,CD,DC,DB,AB\},\{BA,BC,DF,ED,EF\}$ |

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Gong, Z.; Zhang, J.
Chromatic Number of Fuzzy Graphs: Operations, Fuzzy Graph Coloring, and Applications. *Axioms* **2022**, *11*, 697.
https://doi.org/10.3390/axioms11120697

**AMA Style**

Gong Z, Zhang J.
Chromatic Number of Fuzzy Graphs: Operations, Fuzzy Graph Coloring, and Applications. *Axioms*. 2022; 11(12):697.
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**Chicago/Turabian Style**

Gong, Zengtai, and Jing Zhang.
2022. "Chromatic Number of Fuzzy Graphs: Operations, Fuzzy Graph Coloring, and Applications" *Axioms* 11, no. 12: 697.
https://doi.org/10.3390/axioms11120697