Supereulerian graphs in the graph family C2(6, k)

For integers l and k with l>0, and k>=0, C"h(l,k) denotes the collection of h-edge-connected simple graphs G on n vertices such that for every edge-cut X with [email protected]?|X|@?3, each component of G-X has at least (n-k)/l vertices. We prove that for any integer k>0, there exists an integer N=N(k) such that for any n>=N, any graph [email protected]?C"2(6,k) on n vertices is supereulerian if and only if G cannot be contracted to a member in a well-characterized family of graphs. This extends former results in [J. Adv. Math. 28 (1999) 65-69] by Catlin and Li, in [Discrete Appl. Math. 120 (2002) 35-43] by Broersma and Xiong, in [Discrete Appl. Math. 145 (2005) 422-428] by D. Li, Lai and Zhan, and in [Discrete Math. 309 (2009) 2937-2942] by X. Li, D. Li and Lai.

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