Many-to-Many Disjoint Paths in Augmented Cubes With Exponential Fault Edges

It is common knowledge that edge disjoint paths have close relationship with the edge connectivity. Motivated by the well-known Menger theorem, we find that the maximum cardinality of edge disjoint paths connecting any two disjoint connected subgraphs with
$g$
vertices in
$G$
can also define by the minimum modified edge-cut, called the
$g$
-extra edge-connectivity of
$G~(\lambda _{g}(G))$
. It is the cardinality of the minimum set of edges in
$G$
, if such a set exists, whose deletion disconnects
$G$
and leaves every remaining component with at least
$g$
vertices. The
$n$
-dimensional augmented cube
$AQ_{n}$
is a variant of hypercube
$Q_{n}$
. In this paper, we observe that the
$g$
-extra edge-connectivity of the augmented cube for some exponentially large enough
$g$
exists a concentration behavior, for about 72.22 percent values of
$g\leq 2^{n-1}$
, and that the
$g$
-extra edge-connectivity of
$AQ_{n}$
(
$n\geq 3$
) concentrates on
${\lceil \frac {n}{2}\rceil }-1$
special values. Specifically, we prove that the exact value of
$g$
-extra edge-connectivity of augmented cube is a constant
$2\left({{\lceil \frac {n}{2}\rceil }-r}\right)2^{{\lfloor \frac {n}{2}\rfloor }+r}$
for each integer
$2^{{\lfloor \frac {n}{2}\rfloor }+r}-l_{r}\leq g\leq 2^{{\lfloor \frac {n}{2}\rfloor }+r}$
, where
$n\geq 3$
,
$r=1, 2, \ldots, {\lceil \frac {n}{2}\rceil }-1$
and
$l_{r}=\frac {2^{2r+1}-2}{3}$
if
$n$
is odd and
$l_{r}=\frac {2^{2r+2}-4}{3}$
if
$n$
is even. The above upper and lower bounds of
$g$
are sharp. Moreover, we also obtain the exponential edge disjoint paths in
$AQ_{n}$
with edge faults.