Minimum degree, leaf number and traceability

Abstract

Let G be a finite connected graph with minimum degree δ. The leaf number L(G) of G is defined as the maximum number of leaf vertices contained in a spanning tree of G. We prove that if δ ⩾ 1/2 (L(G) + 1), then G is 2-connected. Further, we deduce, for graphs of girth greater than 4, that if δ ⩾ 1/2 (L(G) + 1), then G contains a spanning path. This provides a partial solution to a conjecture of the computer program Graffiti.pc [DeLaViña and Waller, Spanning trees with many leaves and average distance, Electron. J. Combin. 15 (2008), 1–16]. For G claw-free, we show that if δ ⩾ 1/2 (L(G) + 1), then G is Hamiltonian. This again confirms, and even improves, the conjecture of Graffiti.pc for this class of graphs.