s-Vertex Pancyclic Index

A graph G is vertex pancyclic if for each vertex v ∈ V (G), and for each integer k with 3 ≤ k ≤ |V (G)|, G has a k-cycle Ck such that v ∈ V (Ck). 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 K1,3. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K3}, 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 vps(G), is the least nonnegative integer m such that Lm(G) is s-vertex pancyclic. We show 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). Z. Lili · G. Chen Department of Computer Science and Technology, Nanjing University, Nanjing, China Z. Lili College of Computer and Information, Hohai University, Nanjing, China e-mail: [email protected] Y. Shao (B) Arts and Science, Ohio University Southern, Ironton, OH 45638, USA e-mail: [email protected] X. Xu Department of Mathematics, Jiangsu Institute of Education, Nanjing, China J. Zhou Department of Mathematics, Kutztown University, Kutztown, PA 19530, USA