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The $$s$$-Hamiltonian index. (English) Zbl 1223.05166
Summary: For integers $$k,s$$ with 0$$\leqslant k\leqslant s\leqslant |V(G)|-3$$, a graph $$G$$ is called $$s$$-Hamiltonian if the removal of any $$k$$ vertices results in a Hamiltonian graph. For a simple connected graph that is not a path, a cycle or a $$K_{1,3}$$ and an integer $$s\geqslant$$0, we define $$h_s(G)= \min \{m:L^m(G)$$ is $$s$$-Hamiltonian $$\}$$ and $$l(G)=\max\{m:G$$ 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 non-closed path in $$G$$ whose internal vertices have degree 2 in $$G$$. We prove that $$h_s(G)\leqslant l(G)+s+1$$.

##### MSC:
 05C45 Eulerian and Hamiltonian graphs
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##### References:
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