Optimally super-edge-connected transitive graphs

Let X=(V,E) be a connected regular graph. X is said to be super-edge-connected if every minimum edge cut of X is a set of edges incident with some vertex. The restricted edge connectivity λ'(X) of X is the minimum number of edges whose removal disconnects X into non-trivial components. A super-edge-connected k-regular graph is said to be optimally super-edge-connected if its restricted edge connectivity attains the maximum 2k - 2. In this paper, we define the λ'-atoms of graphs with respect to restricted edge connectivity and show that if X is a k-regular k-edge-connected graph whose λ'-atoms have size at least 3, then any two distinct λ'-atoms are disjoint. Using this property, we characterize the super-edge-connected or optimally super-edge -connected transitive graphs and Cayley graphs. In particular, we classify the optimally super-edge -connected quasiminimal Cayley graphs and Cayley graphs of diameter 2. As a consequence, we show that almost all Cayley graphs are optimally super-edge-connected.