Abstract
Let ω ( G ) denote the number of components of a graph G. A connected graph G is said to be 1-tough if ω ( G − X )≤| X | for all X ⊆ V ( G ) with ω ( G − X )>1 . It is well-known that every hamiltonian graph is 1-tough, but that the reverse statement is not true in general, and even not for triangle-free graphs. We present two classes of triangle-free graphs for which the reverse statement holds, i.e., for which hamiltonicity and 1-toughness are equivalent. Our two main results give partial answers to two conjectures due to Nikoghosyan.
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