Randomly twisted hypercubes

Abstract A twisted hypercube of dimension k is created from two twisted hypercubes of dimension k − 1 by adding a matching joining their vertex sets, with the twisted hypercube of dimension 0 consisting of one vertex and no edges. We generate random twisted hypercube by generating the matchings randomly at each step. We show that, asymptotically almost surely, joining any two vertices in a random twisted hypercube of dimension k there are k internally disjoint paths of length at most k lg k + O k lg 2 k . Since the graph is k -regular with 2 k vertices, the number of paths is optimal and the length is asymptotically optimal.

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