The Chvatal-Erdos Condition for a Graph to Have a Spanning Trail

Chvatal and ErdAs proved a well-known result that every graph with connectivity not less than its independence number is Hamiltonian. Han et al. (in Discret Math 310:2082-2090, 2003) showed that every 2-connected graph with is supereulerian with some exceptional graphs. In this paper, we investigate the similar conditions and show that every 2-connected graph with has a spanning trail. We also show that every connected graph with has a spanning trail or is the graph obtained from by replacing at most two vertices of degree 1 in with a complete graph or is the graph obtained from by adding a pendent edge to each vertex of . As a byproduct, we obtain that the line graph of a connected graph with is traceable. These results are all best possible.