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Strong Menger connectedness of augmented -ary -cubes
A connected graph is called strongly Menger (edge) connected if for any
two distinct vertices of , there are vertex(edge)-disjoint paths between and . In this paper, we
consider strong Menger (edge) connectedness of the augmented -ary -cube
, which is a variant of -ary -cube . By exploring the
topological proprieties of , we show that for
(resp.\ for and ) is still strongly Menger
connected even when there are (resp.\ ) faulty vertices and
is still strongly Menger edge connected even when there are
faulty edges for and . Moreover, under the restricted
condition that each vertex has at least two fault-free edges, we show that
is still strongly Menger edge connected even when there are
faulty edges for and . These results are all optimal in the
sense of the maximum number of tolerated vertex (resp.\ edge) faults.Comment: 18 pages, 4 figure