On Hamiltonian Line Graphs

Let \(G\) be a 3-edge-connected simple triangle-free graph of order \(n\) . Using a contraction method, we prove that if \(\delta(G) \geq 4\) and if \(d(u) + d(v) > n/10\) whenever \(uv \in E(G)\) (or whenever \(uv \notin E(G)\) ), then the graph \(G\) has a spanning eulerian sub-
graph. This implies that the line graph \(L(G)\) is hamiltonian. We shall also characterize the extremal graphs.