Abstract
A graph G is vertex pancyclic if for each vertex \({v \in V(G)}\) , and for each integer k with 3 ≤ k ≤ |V(G)|, G has a k-cycle C k such that \({v \in V(C_k)}\) . Let s ≥ 0 be an integer. If the removal of at most s vertices in G results in a vertex pancyclic graph, we say G is an s-vertex pancyclic graph. Let G be a simple connected graph that is not a path, cycle or K 1,3. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K 3}, where a divalent path in G is a path whose interval vertices have degree two in G. The s-vertex pancyclic index of G, written vp s (G), is the least nonnegative integer m such that L m(G) is s-vertex pancyclic. We show that for a given integer s ≥ 0,
And we improve the bound for essentially 3-edge-connected graphs. The lower bound and whether the upper bound is sharp are also discussed.
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Z. Lili was partially supported by the China NSF grants (61003224), China Postdoctoral Science Foundation (20090461087), and the Fundamental Research Funds for the Central Universities.
Y. Shao was partially supported by the NSF-AWM Mentoring Travel Grant.
G. Chen was partially supported by the China NSF grants (60825205) and China 973 project (2006CB303000).
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Lili, Z., Shao, Y., Chen, G. et al. s-Vertex Pancyclic Index. Graphs and Combinatorics 28, 393–406 (2012). https://doi.org/10.1007/s00373-011-1052-z
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DOI: https://doi.org/10.1007/s00373-011-1052-z