Abstract
For integers l and k with l > 0 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 κ ( 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.
Published Version
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