Vertex Connectivity of the Second Iterated Line Graphs with small essential edge connectivity

The line graph of a graph G, denoted by L(G), has E(G) as its vertex set, where two vertices in L(G) are adjacent if and only if the corresponding edges in G have a common vertex. The second iterated line graph of G, written L-2(G), is the line graph of L(G). Let delta(G) be the minimum degree of G. In 2003, Niepel and Knor proved that if G is a 4-connected graph, then the lower bound of the vertex connectivity of L-2(G) is 4 delta(G) - 6. In 2010, it was proved in [4] that the same lower bound can be obtained for essentially 4-edge-connected and 3-connected graphs. In this paper, we determine the lower bounds in terms of the minimum degree for graphs with small essential edge connectivity.