SUFFICIENT CONDITIONS FOR TRIANGLE-FREE GRAPHS TO BE SUPER-λ′

An edge-cut f of a connected graph g is called a restricted edge-cut if g λ′ f containsno isolated vertices. the minimum cardinality of all restricted edge-cuts is called the restricted edge-connectivity λ′(g) of g. a graph g is said to be λ′-optimal if λ′(g) = λ′(g), where ε(g) is the minimumedge-degree of g. a graph is said to be super-λ′if every minimum restricted edge-cut isolates an edge.in this paper, rst, we provide a short proof of a previous theorem about the sufficient conditionfor λ′-optimality in triangle-free graphs, which was given in [j. yuan and a. liu, sufficient conditionsfor λ′k-optimality in triangle-free graphs, discrete math., 310 (2010) 981{987}. second, we generalizea known result about the sufficient condition for triangle-free graphs being super-λ′which was givenby shang et al. in [l. shang and h. p. zhang, sufficient conditions for graphs to be λ′-optimal andsuper-λ′, networks, 309 (2009) 3336{3345].