Estimating the circumference of a graph in terms of its leaf number

Let \(\mathcal {T}\) be the set of spanning trees of a graph G and let L(T) be the number of leaves in a tree T. The leaf number L(G) of G is defined as \(L(G)=\max \{L(T)|T\in \mathcal {T}\}\). Let G be a connected graph of order n and minimum degree \(\delta \) such that \(L(G)\le 2\delta -1\). We show that the circumference of G is at least \(n-1\), and that if G is regular then G is hamiltonian.