Bounds on the connectivity of iterated line graphs - Yehong Shao (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2022)
For simple connected graphs that are neither paths nor cycles, we define l(G)=maxm : G has a divalent path of length m that is not both of length 2 and in a K3, where a divalent path is a path whose internal vertices have degree two in G. Let G be a graph and Ln(G) be its n-th iterated line graph of G. We use κe′(G) and κ(G) for the essential edge connectivity and vertex connectivity of G, respectively. Let G be a simple connected graph that is not a path, a cycle or K1, 3, with l(G)=l ≥ 1. We prove that (i) for integers s ≥ 1, κ′e(Ll + s(G)) ≥ 2s + 2; (ii) for integers s ≥ 2, κ(Ll + s(G)) ≥ 2s − 1 + 2. The bounds are best possible.