Degree sequences and edge connectivity

Abstract

For each positive integer k, we give a finite list C(k) of Bondy–Chvátal type conditions on a nondecreasing sequence $d=(d_1,\dots ,d_n)$ 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 $k=3$, this verifies a conjecture of Bauer, Hakimi, Kahl, and Schmeichel, and for $k=2$ we obtain an alternative proof for their result on bridgeless connected graphs. The explicit list for $k=4$ is given, too.