Rearrangeable Graphs and Rearrangeability Numbers

Abstract

Given a directed graph D(V,A), where V = {1, 2, 3, . . . , n} is the vertex set and A is the directed arc set. A permutation is said to be realizable on D if there exist n arc-disjoint paths in D that connect vertex vi to vertex vi ,i = 1, 2 . . . , n. D is said to be rearrangeable if all n! permutations are realizable on D. Moreover, the rearrangeability number of D is the minimum multiplicity that every arc needs to be duplicated in order for D to become rearrangeable. In this paper, we prove that complete n-partite digraphs are rearrangeable graphs. Then, we give the upper and lower bounds of the rearrangeability number of torus networks.