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.