Reliability assessment for k-ary n-cubes with faulty edges

Abstract

The g-restricted edge connectivity is an important measurement to assess the reliability of networks. The g-restricted edge connectivity of a connected graph G is the minimum size of a set of edges in G, if it exists, whose deletion separates G and leaves every vertex in the remaining components with at least g neighbors. The k-ary n-cube is an extension of the hypercube network and has many desirable properties. It has been used to build the architecture of the Supercomputer Fugaku. This paper establishes that for g≤n, the g-restricted edge connectivity of 3-ary n-cubes is 3⌊g/2⌋(1+(gmod2))(2n−g), and the g-restricted edge connectivity of k-ary n-cubes with k≥4 is 2g(2n−g). These results imply that in Qn3 with at most 3⌊g/2⌋(1+(gmod2))(2n−g)−1 faulty edges, or Qnk(k≥4) with at most 2g(2n−g)−1 faulty edges, if each vertex is incident with at least g fault-free edges, then the remaining network is connected.