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<p><b>Abstract</b>—Twisted cubes are variants of hypercubes. In this paper, we study the optimal embeddings of paths of all possible lengths between two arbitrary distinct nodes in twisted cubes. We use <tmath>TQ_{n}</tmath> to denote the <tmath>n{\hbox{-}}{\rm dimensional}</tmath> twisted cube and use <tmath>{\rm dist}(TQ_{n},u,v)</tmath> to denote the distance between two nodes <tmath>u</tmath> and <tmath>v</tmath> in <tmath>TQ_{n}</tmath>, where <tmath>n \geq 1</tmath> is an odd integer. The original contributions of this paper are as follows: 1) We prove that a path of length <tmath>l</tmath> can be embedded between <tmath>u</tmath> and <tmath>v</tmath> with dilation 1 for any two distinct nodes <tmath>u</tmath> and <tmath>v</tmath> and any integer <tmath>l</tmath> with <tmath>{\rm dist}(TQ_{n},u,v) + 2 \le l\le 2^{n} - 1 (n \geq 3)</tmath> and 2) we find that there exist two nodes <tmath>u</tmath> and <tmath>v</tmath> such that no path of length <tmath>{\rm dist}(TQ_{n},u,v) + 1</tmath> can be embedded between <tmath>u</tmath> and <tmath>v</tmath> with dilation 1 <tmath>(n \geq 3)</tmath>. The special cases for the nonexistence and existence of embeddings of paths between nodes <tmath>u</tmath> and <tmath>v</tmath> and with length <tmath>{\rm dist}(TQ_{n},u,v) + 1</tmath> are also discussed. The embeddings discussed in this paper are optimal in the sense that they have dilation 1.</p>
Twisted cube, interconnection network, path, edge-pancyclicity, embedding, dilation.
Jianxi Fan, Xiaohua Jia, Xiaola Lin, "Optimal Embeddings of Paths with Various Lengths in Twisted Cubes", IEEE Transactions on Parallel & Distributed Systems, vol. 18, no. , pp. 511-521, April 2007, doi:10.1109/TPDS.2007.1003
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