Connectivity, Traceability and Hamiltonicity

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.