Distance and Short Parallel Paths in Augmented Cubes

The augmented cube AQn is recursively denned as follows: It has 2n vertices, each labelled by an n-bit binary string a1a2… an. Define AQ1 = K2. For n ≥ 2, AQn is obtained by taking two copies AQ0n-l and AQ1n-l of AQn-l, with vertex sets V(AQ0n-l) = {Oa2…an with 1b2 …bn iff a2a3… an = b2b3…bn or a2a3 … an = b2b3… bn. It can be defined in several more ways. In addition it is known that AQn 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 AQn(n ≠ 3,4), there exist 2n -1 internally disjoint (X, y)-paths of length ≤ [n/2] +1. It follows that the wide-diameter and fault-diameter of AQn are [n/2] + 1(n ≥ 5). We also determine the average distance and message traffic density in AQn and compare these with those of the hypercube and its variations.