All-fault-tolerant embedding of a complete binary tree in a group of Cayley graphs

Abstract

The paper solves the problems of fault-tolerant embeddings of a complete binary tree in a group of Cayley graphs. First, a complete binary tree (CBT) is embedded into a complete-transposition graph. Then, the derived result is used to further induce the CBT embeddings for the other Cayley graphs. The primary results are that a CBT with height k × (n – 2k + 1) + (k – 2) × 2k + 1, where k = ⌋log n⌊, can be embedded into an n-dimensional complete transposition graph (CTn), star graph (STn) and bubblesort graph (BSn) with dilations 1, 3, and 2n – 3, respectively. Furthermore, a fault-tolerant scheme is developed to recover multiple faults up to the size of the embedded CBT with the least recovery cost. The dilations after recovery become at most 3, 5, and 2n – 1 for the CTn, STn, and BSn, respectively.