Geodesic
Pancyclicity when each Cycle Must Pass Exactly k Hamilton Cycle Chords
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 533-539

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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 each of which passes exactly k chords? For fixed k, we establish a lower bound of Ω(n1/k) on the growth rate.
Keywords: extremal graph theory, pancyclic graph, Hamilton cycle 
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Affif Chaouche, Fatima; Rutherford, Carrie G.; Whitty, Robin W. Pancyclicity when each Cycle Must Pass Exactly k Hamilton Cycle Chords. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 533-539. http://geodesic.mathdoc.fr/item/DMGT_2015_35_3_a10/