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

For integers ll and kk with l>0l>0 and k≥0k≥0, let C(l,k)C(l,k) denote the family of 2-edge-connected graphs GG such that for every bond SS with two or three edges, each component of G−SG−S has at least (|V(G)|−k)/l(|V(G)|−k)/l vertices. In this note we get: (1) If G∈C(6,5)G∈C(6,5) and |V(G)|>35|V(G)|>35, then GG is supereulerian if and only if GG cannot be contracted to some well classified special graphs. (2) If G∈C(6,3)G∈C(6,3), and |V(G)|>21|V(G)|>21, then L(G)L(G), the line graph of GG, is Hamilton-connected if and only if κ(L(G))≥3κ(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.