Abstract
Let G be a simple, connected, triangle-free graph with minimum degree \(\delta \), order n and leaf number L(G). If G has a cut-vertex, we prove that \(L(G)\ge 4\delta -4\) and \(n\ge 4\delta -1\). Both lower bounds are sharp. The lower bound on the leaf number strengthens a result by Mukwembi for triangle-free graphs. As corollaries, we deduce sufficient conditions for connectivity, traceability and Hamiltonicity in triangle-free graphs. As an easy extension of a result by Goodman and Hedetiniemi, we show that a simple, connected, claw-free, paw-free graph G is Hamiltonian if and only if G is not a path. We consider only simple graphs, that is, graphs with neither loops nor multiple edges.
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