A topological property of hypercubes: node disjoint paths

A graph G=(V,E) ( mod V mod >or=2n) satisfies property P/sub n/ if given any 2n distinct vertices s/sub 1/,s/sub 2/, . . ., s/sub n/, g/sub 1/,g/sub 2/, . . ., g/sub n/ in V, there exist n node disjoint paths p/sub i/(s/sub i/, g/sub i/) (i=1, . . ., n) in G such that p/sub i/ is a path from s/sub i/ to g/sub i/. A necessary (but not sufficient) condition for any graph to satisfy P/sub n/ is that it be (2n-1)-connected (MW). The authors prove the optimal result that the hypercube of dimension k, Q(k) (k>or=2n-1), satisfies P/sub n/(n>or=3). The proof is constructive and yields an O(n/sup 3/ log n) algorithm for finding up to (n/2) node disjoint paths in Q(n) (n>or=4) between mutually disjoint source-goal node pairs.<
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