SUFFICIENT CONDITIONS FOR TRIANGLE-FREE GRAPHS TO BE SUPER

An edge-cut F of a connected graph G is called a restricted edge-cut if G F contains no isolated vertices. The minimum cardinality of all restricted edge-cuts is called the restricted edge- connectivity ′ (G) of G. A graph G is said to be ′ -optimal if ′ (G) = (G), where (G) is the minimum edge-degree of G. A graph is said to be super- ′ 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 condition for ′ -optimality in triangle-free graphs, which was given in [J. Yuan and A. Liu, Sufficient conditions for k-optimality in triangle-free graphs, Discrete Math., 310 (2010) 981{987]. Second, we generalize a known result about the sufficient condition for triangle-free graphs being super- ′ which was given by Shang et al. in [L. Shang and H. P. Zhang, Sufficient conditions for graphs to be ′ -optimal and super- ′ , Networks, 309 (2009) 3336{3345].