For each positive integer k, we give a finite list C(k) of Bondy–Chvátal type conditions on a nondecreasing sequence $$d=(d_1,\dots ,d_n)$$d=(d1,⋯,dn) of nonnegative integers such that every graph on n vertices with degree sequence at least d is k-edge-connected. These conditions are best possible in the sense that whenever one of them fails for d then there is a graph on n vertices with degree sequence at least d which is not k-edge-connected. We prove that C(k) is and must be large by showing that it contains p(k) many logically irredundant conditions, where p(k) is the number of partitions of k. Since, in the corresponding classic result on vertex connectivity, one needs just one such condition, this is one of the rare statements where the edge connectivity version is much more difficult than the vertex connectivity version. Furthermore, we demonstrate how to handle other types of edge-connectivity, such as, for example, essential k-edge-connectivity. We prove that any sublist equivalent to C(k) has length at least p(k), where p(k) is the number of partitions of k, which is in contrast to the corresponding classic result on vertex connectivity where one needs just one such condition. Furthermore, we demonstrate how to handle other types of edge-connectivity, such as, for example, essential k-edge-connectivity. Finally, we informally describe a simple and fast procedure which generates the list C(k). Specialized to $$k=3$$k=3, this verifies a conjecture of Bauer, Hakimi, Kahl, and Schmeichel, and for $$k=2$$k=2 we obtain an alternative proof for their result on bridgeless connected graphs. The explicit list for $$k=4$$k=4 is given, too.

中文翻译：

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度数序列和边连通性

对于每个正整数 k，我们在非负序列 $$d=(d_1,\dots ,d_n)$$d=(d1,⋯,dn) 上给出一个 Bondy–Chvátal 类型条件的有限列表 C(k)整数，使得度数序列至少为 d 的 n 个顶点上的每个图都是 k 边连通的。这些条件在某种意义上是最好的，只要它们中的一个对 d 失败，则在 n 个顶点上存在一个度数序列至少为 d 的图，它不是 k 边连接的。我们通过证明 C(k) 包含 p(k) 个逻辑冗余条件来证明 C(k) 是并且必须很大，其中 p(k) 是 k 的分区数。因为，在顶点连通性的相应经典结果中，只需要一个这样的条件，这是边连通性版本比顶点连通性版本困难得多的罕见陈述之一。此外，我们演示了如何处理其他类型的边连接，例如基本的 k 边连接。我们证明任何等价于 C(k) 的子列表的长度至少为 p(k)，其中 p(k) 是 k 的分区数，这与顶点连通性的相应经典结果形成对比，其中一个只需要一个这样的条件。此外，我们演示了如何处理其他类型的边连接，例如基本的 k 边连接。最后，我们非正式地描述了一个生成列表 C(k) 的简单而快速的过程。专门针对 $$k=3$$k=3，这验证了 Bauer、Hakimi、Kahl 和 Schmeichel 的猜想，对于 $$k=2$$k=2，我们获得了他们在无桥连接上的结果的替代证明图表。还给出了 $$k=4$$k=4 的显式列表。