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

Let \(\mathcal {T}\) $\mathcal{T}$ 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 \(L(G)=\max \{L(T)|T\in \mathcal {T}\}\) $L(G)=max\{L(T)|T\in \mathcal{T}\}$ . Let G be a connected graph of order n and minimum degree \(\delta \) $\delta$ such that \(L(G)\le 2\delta -1\) $L(G)\le 2\delta -1$ . We show that the circumference of G is at least \(n-1\) $n-1$ , and that if G is regular then G is hamiltonian.