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

For integers l and k with l > 0, and k ≥ 0, Ch(l, k) denotes the collection of h-edgeconnected simple graphs G on n vertices such that for every edge-cut X with 2 ≤ |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 G ∈ C2(6, k) on n vertices is supereulerian if and only if G cannot be contracted to a member in a wellcharacterized 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. © 2010 Elsevier B.V. All rights reserved.