Lai's Conditions for Spanning and Dominating Closed Trails

A graph is supereulerian if it has a spanning closed trail. For an integer r, let Q0(r) be the family of 3-edge-connected nonsupereulerian graphs of order at most r. For a graph G, define δL(G) = min{max{d(u), d(v)}| for any uv ∈ E(G)}. For a given integer p > 2 and a given real number , a graph G of order n is said to satisfy a Lai’s condition if δL(G) > np − . In this paper, we show that if G is a 3-edgeconnected graph of order n with δL(G) > np − , then there is an integer N(p, ) such that when n > N(p, ), G is supereulerian if and only if G is not a graph obtained from a graph Gp in the finite family Q0(3p−5) by replacing some vertices in Gp with nontrivial graphs. Results on the best possible Lai’s conditions for Hamiltonian line graphs of 3-edge-connected graphs or 3-edge-connected supereulerian graphs are given, which are improvements of the results in [J. Graph Theory 42(2003) 308-319] and in [Discrete Mathematics, 310(2010) 2455-2459].