# Path Embeddings with Prescribed Edge in the Balanced Hypercube Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- 1.
- $(({a}_{0}+1)\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4,{a}_{1},\dots ,{a}_{i-1},{a}_{i},{a}_{i+1},\dots ,{a}_{n-1})$,$(({a}_{0}-1)\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4,{a}_{1},\dots ,{a}_{i-1},{a}_{i},{a}_{i+1},\dots ,{a}_{n-1})$, and
- 2.
- $(({a}_{0}+1)\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4,{a}_{1},\dots ,{a}_{i-1},({a}_{i}+{(-1)}^{{a}_{0}})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4,{a}_{i+1},\dots ,{a}_{n-1})$,$(({a}_{0}-1)\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4,{a}_{1},\dots ,{a}_{i-1},({a}_{i}+{(-1)}^{{a}_{0}})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4,{a}_{i+1},\dots ,{a}_{n-1})$.

**Proposition**

**3.**

## 3. Main Results

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**4.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Case**

**1:**

**Case**

**2:**

**Theorem**

**1.**

**Proof.**

**Case**

**1:**

**Case**

**2:**

**Case**

**3:**

**Case**

**4:**

**Case**

**5:**

**Case**

**6:**

**Case**

**7:**

**Case**

**8:**

**Case**

**9:**

**Case**

**10:**

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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x | y | Hamiltonian Paths Passing through e with Neither x nor y Being Incident to e | |
---|---|---|---|

(1) | (3,0) | (2,0) | (3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1)(3,1)(0,1)(1,1)(0,0)(1,0)(2,0) |

(2) | (3,0) | (0,1) | (3,0)(0,0)(1,0)(2,3)(3,3)(0,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1)(3,1)(2,0)(1,1)(0,1) |

(3) | (3,0) | (2,2) | (3,0)(0,3)(3,3)(2,3)(1,0)(0,0)(3,1)(2,0)(1,1)(0,1)(1,2)(2,1)(3,2)(0,2)(1,3)(2,2) |

(4) | (3,0) | (0,3) | (3,0)(0,0)(1,0)(2,0)(3,1)(0,1)(1,1)(2,1)(1,2)(2,2)(3,2)(0,2)(1,3)(2,3)(3,3)(0,3) |

(5) | (1,1) | (2,1) | (1,1)(0,1)(3,1)(2,0)(1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1) |

(6) | (1,1) | (2,2) | (1,1)(0,1)(3,1)(2,0)(1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,1)(1,2)(2,2) |

(7) | (1,1) | (2,3) | (1,1)(0,0)(3,1)(0,1)(1,2)(2,1)(3,2)(2,2)(1,3)(0,2)(3,3)(0,3)(1,0)(2,0)(3,0)(2,3) |

(8) | (1,2) | (2,2) | (1,2)(2,1)(1,1)(0,1)(3,1)(2,0)(1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2) |

(9) | (1,2) | (2,3) | (1,2)(2,1)(1,1)(0,1)(3,1)(2,0)(1,0)(0,0)(3,0)(0,3)(1,3)(2,2)(3,2)(0,2)(3,3)(2,3) |

(10) | (1,3) | (2,3) | (1,3)(0,3)(3,0)(0,0)(1,0)(2,0)(1,1)(2,1)(3,1)(0,1)(3,2)(2,2)(1,2)(0,2)(3,3)(2,3) |

x | y | Hamiltonian Paths Passing through e with x or y Being Incident to e | |
---|---|---|---|

(1) | (1,0) | (2,0) | (1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1)(1,1)(0,1)(3,1)(2,0) |

(2) | (1,0) | (0,1) | (1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1)(1,1)(2,0)(3,1)(0,1) |

(3) | (1,0) | (0,2) | (1,0)(0,0)(3,0)(0,3)(1,3)(2,3)(3,3)(2,2)(3,2)(2,1)(1,1)(2,0)(3,1)(0,1)(1,2)(0,2) |

(4) | (1,0) | (0,3) | (1,0)(0,0)(3,0)(2,0)(3,1)(0,1)(1,1)(2,1)(1,2)(2,2)(3,2)(0,2)(1,3)(2,3)(3,3)(0,3) |

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**MDPI and ACS Style**

Chen, D.; Lu, Z.; Shen, Z.; Zhang, G.; Chen, C.; Zhou, Q.
Path Embeddings with Prescribed Edge in the Balanced Hypercube Network. *Symmetry* **2017**, *9*, 79.
https://doi.org/10.3390/sym9060079

**AMA Style**

Chen D, Lu Z, Shen Z, Zhang G, Chen C, Zhou Q.
Path Embeddings with Prescribed Edge in the Balanced Hypercube Network. *Symmetry*. 2017; 9(6):79.
https://doi.org/10.3390/sym9060079

**Chicago/Turabian Style**

Chen, Dan, Zhongzhou Lu, Zebang Shen, Gaofeng Zhang, Chong Chen, and Qingguo Zhou.
2017. "Path Embeddings with Prescribed Edge in the Balanced Hypercube Network" *Symmetry* 9, no. 6: 79.
https://doi.org/10.3390/sym9060079