Eulerian subgraphs and Hamilton-connected line graphs

Let C(l, k) denote a class of 2-edge-connected graphs of order n such that a graph G ∈ C(l, k) if and only if for every edge cut S ⊆ E(G) with |S| ≤ 3, each component of G − S has order at least n− k l . We prove the following. (1) If G ∈ C(6, 0), then G is supereulerian if and only if G cannot be contracted to K2,3, K2,5 or K2,3(e), where e ∈ E(K2,3) and K2,3(e) stands for a graph obtained from K2,3 by replacing e by a path of length 2. (2) If G ∈ C(6, 0) and n ≥ 7, then L(G) is Hamilton-connected if and only if κ(L(G)) ≥ 3. Former results by Catlin and Li, and by Broersma and Xiong are extended.