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It is known that Θ(log n) chords must be added to an n-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, Θ(n) chords are required. A possibly 'intermediate' variation is the following: given k, 1 ≤ k ≤ n, how many chords must be added to ensure that there exist cycles of every possible length… (More)

Let (E, A) be a set system consisting of a finite collection A of subsets of a ground set E, and suppose that we have a function φ which maps A into some set S. Now removing a subset K from E gives a restriction A(¯ K) to those sets of A disjoint from K, and we have a corresponding restriction φ| A(¯ K) of our function φ. If the removal of K does not affect… (More)

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