Connectivity of iterated line graphs

Let k≥0k≥0 be an integer and Lk(G)Lk(G) be the kkth iterated line graph of a graph GG. Niepel and Knor proved that if GG is a 4-connected graph, then κ(L2(G))≥4δ(G)−6κ(L2(G))≥4δ(G)−6. We show that the connectivity of GG can be relaxed. In fact, we prove in this note that if GG is an essentially 4-edge-connected and 3-connected graph, then κ(L2(G))≥4δ(G)−6κ(L2(G))≥4δ(G)−6. Similar bounds are obtained for essentially 4-edge-connected and 2-connected (1-connected) graphs.