On hamiltonicity of 1-tough triangle-free graphs

− ≤ | | ⊆ − 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.