Unpaired disjoint path covers in bipartite torus-like graphs with edge faults

Abstract

One of the essential problems in parallel processing is finding disjoint paths in the graphs representing interconnection networks. Regarding the disjoint paths, it is often needed to discover a disjoint path cover in a graph, which is a set of pairwise vertex-disjoint paths containing every vertex. A special case of the disjoint path cover is the unpaired (many-to-many) k-disjoint path cover $\{ P_1, \ldots , P_k\}$, where, given two disjoint vertex subsets S and T each of size k, every mutually disjoint path $P_i$ connects a vertex of S to another in T. In this paper, we find that if a bipartite torus-like graph is built from lower dimensional torus-like graphs having some decent properties on unpaired disjoint path cover, the new graph inherits such properties. Utilizing this result, we show that, for all $m \ge 2$, $f\ge 0$, and $k \ge 1$ such that $f+k \le 2m-1$, an m-dimensional bipartite torus with at most f edge faults has an unpaired k-disjoint path cover joining two size-k sets S and T each from different bipartition subsets of vertices. The upper bound $2m-1$ on $f+k$ is the best possible.