ABSTRACT

A ${\lambda }_m$-connected graph G is optimally m-restricted edge-connected if its m-restricted edge-connectivity ${\lambda }_m\left(G\right)$ equals the minimum m-edge-degree ${\xi }_m\left(G\right)$, and super m-restricted edge-connected if all minimum m-restricted edge cuts are trivial. In this study, a bound for the size of graphs with girth greater than any given integer l $(l\ge 3)$ is obtained firstly. Then the second minimum degree conditions for optimally and super m-restricted edge-connected graphs with girth greater than l are provided. Furthermore, the relationship between optimally m-restricted edge connectivity and optimally $(m+1)$-restricted edge connectivity is characterized.

中文翻译：

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给定周长的最优和超 m 限制边连通图的充分条件

摘要

一种${\lambda }_{米}$连通图G是最优m限制边连通的，如果它的m限制边连通性${\lambda }_{米}\left(G\right)$等于最小m -edge-degree${\xi }_{米}\left(G\right)$, 如果所有最小m限制边缘切割都是微不足道的，则超级m限制边缘连接。在这项研究中，周长大于任何给定整数l的图大小的界限 $(l\ge 3)$首先获得。然后给出了周长大于l的最优超m限制边连通图的第二最小度条件。此外，最优m限制边连通性与最优$(米+1)$- 限制边缘连接的特征。