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Edge-connectivity and edge-disjoint spanning trees. (English) Zbl 1168.05039
For a subset $$X$$ of edges of a graph $$G$$, $$\omega(G-X)$$ denotes the number of components of the subgraph $$G-X$$. For an integer $$c$$ such that $$2\leq c\leq|V(G)|$$ the higher order of edge-connectivity, $$\lambda_c(G)$$, and the higher order of edge-toughness, $$\tau_{c-1}(G)$$, are defined by $$\lambda_c(G)=\min\{|X|\}$$, $$\tau_{c-1}(G)=\min\frac{|X|}{\omega(G-X)-c}$$, where the minima are taken over all subsets $$X\subseteq E(G)$$ such that $$\omega(G-X)\geq c$$ [C. C. Chen, K. M. Koh and Y. H. Peng, “On the higher-order edge toughness of a graph”, Discrete Math. 111, No. 1–3, 113–123 (1993; Zbl 0789.05086)].
From the authors’ introduction and abstract: “Over ten years ago Catlin left an unpublished note proving a theorem which characterizes the edge-connectivity of a connected graph $$G$$ in terms of the spanning tree packing numbers of its subgraphs. …Sadly the author passed away on April 20, 1995. …In this paper we establish a relationship between $$\lambda_c(G)$$ and $$\tau_{c-1}(G)$$ which gives a characterization of the edge-connectivity of a graph $$G$$ in terms of the spanning tree packing number of subgraphs of $$G$$. The digraph analogue is also obtained. The main results are applied to show that, if a graph $$G$$ is $$s$$-hamiltonian, then $$L(G)$$ is also $$s$$-hamiltonian, and that, if a graph $$G$$ is $$s$$-hamiltonian-connected, then $$L(G)$$ is also $$s$$-hamiltonian-connected.”

##### MSC:
 05C40 Connectivity 05C05 Trees 05C45 Eulerian and Hamiltonian graphs 05C20 Directed graphs (digraphs), tournaments
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##### References:
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