Unpaired Many-to-Many Disjoint Path Covers in Nonbipartite Torus-Like Graphs With Faulty Elements

One of the key problems in parallel processing is finding disjoint paths in the underlying graph of an interconnection network. The disjoint path cover of a graph is a set of pairwise vertex-disjoint paths that altogether cover every vertex of the graph. Given disjoint source and sink sets,
$S = \{ s_{1},\ldots,s_{k}\}$
and
$T = \{t_{1},\ldots, t_{k}\}$
, in graph
$G$
, an unpaired many-to-many
$k$
-disjoint path cover joining
$S$
and
$T$
is a disjoint path cover
$\{ P_{1}, \ldots, P_{k}\}$
, in which each path
$P_{i}$
runs from source
$s_{i}$
to some sink
$t_{j}$
. In this paper, we reveal that a nonbipartite torus-like graph, if built from lower dimensional torus-like graphs that have good disjoint-path-cover properties of the unpaired type, retains such a good property. As a result, an
$m$
-dimensional nonbipartite torus,
$m \geq 2$
, with at most
$f$
vertex and/or edge faults has an unpaired many-to-many
$k$
-disjoint path cover joining arbitrary disjoint sets
$S$
and
$T$
of size
$k$
each, subject to
$k \geq 1$
and
$f+k \leq 2m-2$
. The bound of
$2m-2$
on
$f+k$
is nearly optimal.