Distance and Short Parallel Paths in Augmented Cubes

Abstract

Abstract The augmented cube AQ n is recursively denned as follows: It has 2 n vertices, each labelled by an n-bit binary string a1a2… an. Define AQ 1 = K 2 . For n ≥ 2, AQ n is obtained by taking two copies AQ 0 n-l and AQ 1 n-l of AQ n-l , with vertex sets V ( AQ 0 n - l ) = {Oa 2 …a n with 1b 2 …b n iff a 2 a 3 … a n = b 2 b 3 …b n or a 2 a 3 … a n = b 2 b 3 … b n . It can be defined in several more ways. In addition it is known that AQ n is a Cayley graph, (2n − l )-regular, (2n − l )-connected (n ≥ 3) and that it has diameter [n/2]. These cubes admit routing and broadcasting procedures that are as simple as those of hypercubes. In this paper, we show that between any two vertices X, Y of AQ n (n ≠ 3,4), there exist 2 n -1 internally disjoint ( X, y )-paths of length ≤ [n/2] +1. It follows that the wide-diameter and fault-diameter of AQ n are [n/2] + 1(n ≥ 5). We also determine the average distance and message traffic density in AQ n and compare these with those of the hypercube and its variations.