Supereulerian graphs in the graph family \(C_{2}(6,k)\) (Q628342)

For integers \(l\) and \(k\) with \(l>0\) and \(k\geq 0\), \(C_h(l,k)\) denotes the collection of \(h\)-edge-connected simple graphs \(G\) on \(n\) vertices such that for every edge-cut \(X\) with \(2 \leq |X| \leq 3\) each component of \(G - X\) has at least \((n - k)/l\) vertices. The authors prove the following result: Theorem. Let \(k>0\) be an integer, then there exists an integer \(N(k) \leq 7k\) such that, for any graph \(G \in C_{2}(6,k)\) with \(|V(G)| > N(k)\), \(G\) is supereulerian if and only if \(G\) cannot be contracted to a member in \({\mathcal{F}}'\). This extends former results of \textit{P. A. Catlin} and \textit{X. Li} [``Super-Eulerian graphs of minimum degree at least 4,'' Adv. Math., Beijing 28, No.\,1, 65--70 (1999; Zbl 1054.05506)], of \textit{H. J. Broersma} and \textit{L. Xiong} [``A note on minimum degree conditions for supereulerian graphs,'' Discrete Appl. Math. 120, No.\,1-3, 35--43 (2002; Zbl 0993.05097)], of \textit{D. Li, H.-J. Lai} and \textit{M. Zhan} [``Eulerian subgraphs and Hamilton-connected line graphs,'' Discrete Appl. Math. 145, No.\,3, 422--428 (2005; Zbl 1057.05053)], and of \textit{X. Li}, \textit{D. Li}, and \textit{H.-J. Lai} [``The supereulerian graphs in the graph family \(C(l,k)\),'' Discrete Math. 309, No.\,9, 2937--2942 (2009; Zbl 1200.05126)].