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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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