an:06118562
Zbl 1256.05125
Lili, Zhang; Shao, Yehong; Chen, Guihai; Xu, Xinping; Zhou, Ju
\(s\)-vertex pancyclic index
EN
Graphs Comb. 28, No. 3, 393-406 (2012).
00299461
2012
j
05C38 05C76
vertex pancyclic graph; vertex pancyclic index; triangular graph; line graph
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.