Traceability of line graphs

Brualdi and Shanny [R.A. Brualdi, R.F. Shanny, Hamiltonian line graphs, J. Graph Theory 5 (1981) 307–314], Clark [L. Clark, On hamitonian line graphs, J. Graph Theory 8 (1984) 303–307] and Veldman [H.J. Veldman, On dominating and spanning circuits in graphs, Discrete Math. 124 (1994) 229–239] gave minimum degree conditions of a line graph guaranteeing the line graph to be hamiltonian. In this paper, we investigate the similar conditions guaranteeing a line graph to be traceable. In particular, we show the following result: let GG be a simple graph of order nn and L(G)L(G) its line graph. If nn is sufficiently large and, either δ(L(G))>2⌊n−84⌋; or δ(L(G))>2⌊n−2010⌋ and GG is almost bridgeless, then L(G)L(G) is traceable. As a byproduct, we also show that every 2-edge-connected triangle-free simple graph with order at most 9 has a spanning trail. These results are all best possible.