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Sufficient conditions for triangle-free graphs to be optimally restricted edge-connected

机译：充分限制无三角形图成为最佳受限边连接的条件

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摘要

For a connected graph G, an edge set S is a k-restricted edge-cut if G-S is disconnected and every component of G - S has at least k vertices. Graphs that allow k-restricted edge-cuts are called λ_k-connected. The k-edge-degree of a graph G is the minimum number of edges between a connected subgraph H of order k and its complement G - H. A λ_k-connected graph is called λ_k-optimal if its k-restricted edge-connectivity equals its minimum k-edgedegree and super-λ_k if every minimum k-restricted edge-cut isolates a connected subgraph of order k. In this paper we consider the cases k = 2 and k = 3. For triangle-free graphs that are not λ_k-optimal, we establish lower bounds for the order of components left by a minimum k-restricted edge-cut in terms of the minimum k-edge-degree. Sufficient conditions for a triangle-free graph to be λ_k-optimal and super-λ_k follow.
机译：对于连接的图G，如果断开G-S的连接且G-S的每个分量至少具有k个顶点，则边集S为k限制的边切割。允许进行k限制切边的图形称为λ_k-connected。图G的k边缘度是k阶连通子图H及其补G-H之间的最小边数。如果λ_k连通图的k约束边连通性等于，则称λ_k连通图。如果每个最小k限制的边切割都隔离了k阶连通子图，则其最小k边缘度和super-λ_k。在本文中，我们考虑了k = 2和k = 3的情况。对于不是λ_k最优的无三角形图，我们为最小k限制的边沿切割所留下的分量的阶数确定了下界。最小k边缘度。满足无三角形图为λ_k最优和超λ_k的充分条件。

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