Abstract
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 G be a simple graph of order n and L ( G ) its line graph. If n is sufficiently large and, either δ ( L ( G ) ) > 2 ⌊ n − 8 4 ⌋ ; or δ ( L ( G ) ) > 2 ⌊ n − 20 10 ⌋ and G is almost bridgeless, then 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.
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