Hamiltonian cycles and paths in hypercubes with disjoint faulty edges
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An n-dimensional hypercube Q_n, is an undirected graph with 2^n nodes, each labeled with a distinct binary string of length n. Two vertices are connected by an edge if their labels differ in one position. The hypercube is bipartite, the set of nodes is the union of two sets: nodes of parity 0 and nodes of parity 1. For any dimension i, there is a partition of Q_n into Q^i_0={x∈ Q_n:x_i=0} and Q^i_1={x∈ Q_n:x_i=1}. For any edge e=(u,v), if u∈ Q^i_0 and v∈ Q^i_1, then we say that e is crossing and goes in i dimension. We say that e is of parity 0 (or 1) if u is of parity 0 (or 1). We consider hypercubes with pairwise disjoint faulty edges. We show that Q_n with n> 4, is Hamiltonian if and only if for each dimension i, there is a healthy crossing edge of parity 0 and a healthy crossing edge of parity 1. Moreover Q_n is laceable (every two vertices of different parity can be connected by a Hamiltonian path) if and only if for each dimension i, there are at least three healthy edges not of the same parity.
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