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The k-Restricted Edge Connectivity of Balanced Bipartite Graphs

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Abstract

For a connected graph G = (V, E), an edge set \({S\subset E}\) is called a k-restricted edge cut if GS is disconnected and every component of GS contains at least k vertices. The k-restricted edge connectivity of G, denoted by λ k (G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex sets \({U_1,U_2\subset V(G)}\), denote the set of edges of G with one end in U 1 and the other in U 2 by [U 1, U 2]. Define \({\xi_k(G)=\min\{|[U,V(G){\setminus} U]|: U}\) is a vertex subset of order k of G and the subgraph induced by U is connected}. A graph G is said to be λ k -optimal if λ k (G) = ξ k (G). A graph is said to be super-λ k if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. In this paper, we present some degree-sum conditions for balanced bipartite graphs to be λ k -optimal or super-λ k . Moreover, we demonstrate that our results are best possible.

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Correspondence to Jun Yuan.

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Yuan, J., Liu, A. The k-Restricted Edge Connectivity of Balanced Bipartite Graphs. Graphs and Combinatorics 27, 289–303 (2011). https://doi.org/10.1007/s00373-010-0966-1

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  • DOI: https://doi.org/10.1007/s00373-010-0966-1

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