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A-connected graph  G  is optimally  m -restricted edge-connected if its  m -restricted edge-connectivityequals the minimum  m -edge-degree, 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  lis 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-restricted edge connectivity is characterized.

Sufficient conditions for optimally and super m -restricted edge-connected graphs with given girth