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 $g\le n$, the g-restricted edge connectivity of 3-ary n-cubes is $3^{\lfloor g/2\rfloor }(1+(g\text{mod}2\left)\right)(2n-g)$, and the g-restricted edge connectivity of k-ary n-cubes with $k\ge 4$ is $2^g(2n-g)$. These results imply that in $Q_n^3$ with at most $3^{\lfloor g/2\rfloor }(1+(g\text{mod}2\left)\right)(2n-g)-1$ faulty edges, or $Q_n^k(k\ge 4)$ with at most $2^g(2n-g)-1$ faulty edges, if each vertex is incident with at least g fault-free edges, then the remaining network is connected.