https://hal.inria.fr/hal-00988176Gregor, PetrPetrGregorDepartment of Theoretical Computer Science and Mathematical Logic [Prague] - CU - Charles University [Prague]Škrekovski, RisteRisteŠkrekovskiOddelek za Matematiko - University of Ljubljana Departement of Mathematics [Slovenia] - University of Ljubljana Long cycles in hypercubes with distant faulty verticesHAL CCSD2009[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Inria Sophia Antipolis-Méditerranée / I3s, Service Ist2014-05-07 16:03:562021-11-10 17:38:032014-05-07 16:05:01enJournal articleshttps://hal.inria.fr/hal-00988176/document10.46298/dmtcs.466application/pdf1In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.