Spanning paths and cycles in triangle-free graphs

Let G be a simple, connected, triangle-free graph with minimum degree delta and leaf number L(G). We prove that if L(G) <= 2 delta - 1, then G is either Hamiltonian or G is an element of F-2, where F-2 is the class of non-Hamiltonian graphs with leaf number 2 delta - 1. Further, if L(G) <= 2 delta, we show that G is traceable or G is an element of F-3. The results, apart from strengthening theorems in [17, 16] for this class of graphs, provide a sucient condition for a triangle-free graph to be Hamiltonian or traceable based on leaf number and minimum degree.