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**The spined cube: a new hypercube variant with smaller diameter.**
*(English)*
Zbl 1260.68027

Summary: Bijective connection graphs (in brief, BC graphs) are a family of hypercube variants, which contains hypercubes, twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes, etc. It was proved that the smallest diameter of all the known \(n\)-dimensional bijective connection graphs (BC graphs) is \(\lceil\frac{n+1}{2}\rceil\), given a fixed dimension \(n\). An important question about the smallest diameter among all the BC graphs is: Does there exist a BC graph whose diameter is less than the known BC graphs such as crossed cubes, twisted cubes, Möbius cubes, etc., with the same dimension? This paper answers this question. In this paper, we find that there exists a kind of BC graphs called spined cubes and we prove that the \(n\)-dimensional spined cube has the diameter \(\lceil n/3\rceil +3\) for any integer \(n\) with \(n\geq 14\). It is the first time in the literature that a hypercube variant with such a small diameter is presented.

### MSC:

68M10 | Network design and communication in computer systems |

68M07 | Mathematical problems of computer architecture |

68R10 | Graph theory (including graph drawing) in computer science |

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\textit{W. Zhou} et al., Inf. Process. Lett. 111, No. 12, 561--567 (2011; Zbl 1260.68027)

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