Note: The supereulerian graphs in the graph family C(l,k)

For integers l and k with l0 and k=0, let C(l,k) denote the family of 2-edge-connected graphs G such that for every bond S with two or three edges, each component of G-S has at least (|V(G)|-k)/l vertices. In this note we get: (1) If G@?C(6,5) and |V(G)|35, then G is supereulerian if and only if G cannot be contracted to some well classified special graphs. (2) If G@?C(6,3), and |V(G)|21, then L(G), the line graph of G, is Hamilton-connected if and only if @k(L(G))=3. Our results extend some earlier results in [P.A. Catlin, X.W. Li, Supereulerian graphs of minimum degree at least 4, J. Adv. Math. 28 (1999) 65-69], [H.J. Broersma, L.M. Xiong, A note on minimum degree conditions for supereulerian graphs, Discrete Appl. Math. 120 (2002) 35-43] and [D.X. Li, H.-J. Lai, M.Q. Zhan, Eulerian subgraphs and hamilton-connected line graphs, Discrete Appl. Math. 145 (2005) 422-428] by Catlin and Li, by Broersma and Xiong, and by Li, Lai and Zhan.