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adjacent algorithm aquarium-keeper route assume binary block colour schemes column Combinatorial competition graphs complete components Computer configuration connected consider construct contains contradiction Corollary corresponding coset cycle of length database defined definition degree denote digraph disjoint edge labeled elements example exists Figure finite follows function given graph G Graph Theory hamiltonian cycle Hence hypercube induced subgraph integer interval graphs Latin square least Lemma Let G linear loop lower bound Math matrix matroid maximal cliques maximum number minimum nodes nonsimplicial number of edges obtained open set optimal orbits pairs partition paths of length permutation planar graph plane points polygon prime primitive roots Proof reliability result route problem satisfies sequenced basis shortest path Subcase subgraph subset Suppose Theorem tournament tree triangle triangle-free graph Type III vertices vertex set vertical edge Vizing's conjecture zoo-keeper route