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$$s$$-vertex pancyclic index. (English) Zbl 1256.05125
Summary: A graph $$G$$ is vertex pancyclic if for each vertex $$v \in V(G)$$, and for each integer $$k$$ with $$3 \leq k \leq |V(G)|$$, $$G$$ has a $$k$$-cycle $$C_{k}$$ such that $$v \in V(C_k)$$. Let $$s \geq 0$$ 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 $$s \geq 0$$,
$vp_s(G) \leq \begin{cases} l(G)+s+1: \quad &\text{if }0 \leq s \leq 4 \\ l(G)+\lceil {\log}_2(s-2) \rceil+4: \quad &\text{if }s \geq 5 \end{cases}.$ And we improve the bound for essentially 3-edge-connected graphs. The lower bound and whether the upper bound is sharp are also discussed.

##### MSC:
 05C38 Paths and cycles 05C76 Graph operations (line graphs, products, etc.)
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