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A fault-free path in the n-dimensional hypercube Qn with f faulty vertices is said to be long if it has length at least 2 n − 2f − 2. Similarly, a fault-free cycle in Qn is long if it has length at least 2 n − 2f. If all faulty vertices are from the same bipartite class of Qn, such length is the best possible. We show that for every set of at most 2n − 4… (More)

Let G k n be the subgraph of the hypercube Q n induced by levels between k and n − k, where n ≥ 2k + 1 is odd. The well-known middle level conjecture asserts that G k 2k+1 is Hamiltonian for all k ≥ 1. We study this problem in G k n for fixed k. It is known that G 0 n and G 1 n are Hamiltonian for all odd n ≥ 3. In this paper we prove that also G 2 n is… (More)

In this paper, we study long cycles in induced subgraphs of hyper-cubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced sub-graph of Q n with minimum degree n − 1 contains a cycle of length at least 2 n − 2f where f is the number of removed vertices. This length is the best… (More)

Given a family {ui, vi} k i=1 of pairwise distinct vertices of the n-dimensional hypercube Qn such that the distance of ui and vi is odd and k ≤ n − 1, there exists a family {Pi} k i=1 of paths such that ui and vi are the endvertices of Pi and {V (Pi)} k i=1 partitions V (Qn). This holds for any n ≥ 2 with one exception in the case when n = k + 1 = 4. On… (More)