La Matematica _#####################_ https://doi.org/10.1007/s44007-021-00014-7 ORIGINAL RESEARCH ARTICLE Connectivity, Traceability and Hamiltonicity P. Mafuta 1 · J. P. Mazorodze 2 · S. Munyira 2 · J. Mushanyu 2 Received: 14 April 2021 / Revised: 10 September 2021 / Accepted: 15 November 2021 © The Author(s), under exclusive licence to Springer Science+Business Media LLC, part of Springer Nature 2022 Abstract Let G be a simple, connected, triangle-free graph with minimum degree δ, order n and leaf number L (G). If G has a cut-vertex, we prove that L (G) ≥ 4δ − 4 and n ≥ 4δ − 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. Keywords Leaf number · Minimum degree · Order · Connectivity · Cycles · Paths Mathematics Subject Classification 05C38 · 05C40 · 05C45 1 Introduction Let G = (V , E ) be a connected graph with vertex set V (G) and edge set E (G). The connectivity κ(G) of G refers to the minimum number of vertices whose deletion B P. Mafuta phillipmafuta@gmail.com J. P. Mazorodze mazorodzejaya@gmail.com S. Munyira munyirask@gmail.com J. Mushanyu mushanyuj@gmail.com 1 Department of Mathematics and Applied Mathematics IB74, University of the Free State, Bloemfontein, South Africa 2 Department of Mathematics and Computational Sciences, University of Zimbabwe, Harare, Zimbabwe 0123456789().: V,-vol 123