A sufficient condition for equality of edge‐connectivity and minimum degree of a graph

Let G be a connected graph of order p ≥ 2, with edge‐connectivity κ 1 ( G ) and minimum degree δ( G ). It is shown her ethat in order to obtain the equality κ 1 ( G ) = δ( G ), it is sufficient that, for each vertex x of minimum degree in G , the vertices in the neighbourhood N ( x ) of x have sufficiently large degree sum. This result implies a previous result of Chartrand, which required that δ( G ) ≥ [ p /2].