Edge degree conditions for dominating and spanning closed trails

Edge degree conditions have been studied since the 1980s, mostly with regard to hamiltonicity of line graphs and the equivalent existence of dominating closed trails in their root graphs, as well as the stronger property of being supereulerian, i.e., admitting a spanning closed trail. For a graph G, let (sigma) over bar (2)(G) = min{d(u) + d(v) vertical bar uv is an element of E(G)}. Chen et al. conjectured that a 3-edge-connected graph G with sufficientl large order n and (sigma) over bar (2)(G) > n/9 - 2 is either supereulerian or contractible to the Petersen graph. We show that the conjecture is true when (sigma) over bar (2)(G) >= 2(left perpendicularn/15right perpendicular - 1). Furthermore, we show that for an essentially k-edge-connected graph G with sufficiently large order n, the following statements hold. (i) If k = 2 and (sigma) over bar (2)(G) >= 2(left perpendicularn/8right perpendicular - 1), then either L(G) is hamiltonian or G can be contracted to one of a set of six graphs which are not supereulerian; (ii) If k = 3 and (sigma) over bar (2)(G) >= 2(left perpendicularn/15right perpendicular = 1), then either L(G) is hamiltonian or G can be contracted to the Petersen graph.