k-restricted edge-connectivity in triangle-free graphs

Let G be a lambda(k)-connected graph. G is called lambda(k)-optimal, if its k-restricted edge-connectivity lambda(k)(G) equals its minimum k-edge degree. G is called super-lambda(k) if every lambda(k)-cut isolates a connected subgraph of order k. Firstly, we give a lower bound on the order of 2-fragments in triangle-free graphs that are not lambda(2)-optimal. Secondly, we present an Ore-type condition for triangle-free graphs to be lambda(3)-optimal. Thirdly, we prove a lower bound on the order of k-fragments in triangle-free lambda(k)-connected graphs, and use it to show that triangle-free graphs with high minimum degree are lambda(k)-optimal and super-lambda(k). (C) 2012 Elsevier B.V. All rights reserved.