Extremality and Sharp Bounds for the k-edge-connectivity of Graphs
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Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named k-edge-connectivity. For any integer k with 2≤ k≤ n, the k-edge-connectivity of a graph G, denoted by λ_k(G), is defined as the smallest number of edges whose removal from G produces a graph with at least k components. In this paper, we first compute some exact values and sharp bounds for λ_k(G) in terms of n and k. We then discuss the relationships between λ_k(G) and other generalized connectivities. An algorithm in O(n^2) time will be provided such that we can get a sharp upper bound in terms of the maximum degree. Among our results, we also compute some exact values and sharp bounds for the function f(n,k,t) which is defined as the minimum size of a connected graph G with order n and λ_k(G)=t.
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