On the extended Clark-Wormold Hamiltonian-like index problem

For a hamiltonian property P, Clark and Wormold introduced the problem of investigating the value P(a,b)=max⁡{min⁡{n:Ln(G) has property P}: κ′(G)≥a and δ(G)≥b}, and proposed a few problems to determine P(a,b) with b≥a≥4 when P is being hamiltonian, edge-hamiltonian and hamiltonian-connected. Zhan in 1986 proved that the line graph of a 4-edge-connected graph is Hamilton-connected, which implies a solution to the unsettled cases of above-mentioned problem. We consider an extended version of the problem. Let ess′(G) denote the essential edge-connectivity of a graph G, and define P′(a,b)=max⁡{min⁡{n:Ln(G) has property P}: ess′(G)≥a and δ(G)≥b}. We investigate the values of P′(a,b) when P is one of these hamiltonian properties. In particular, we show that for any values of b≥1, P′(4,b)≤2 and P′(4,b)=1 if and only if Thomassen's conjecture that every 4-connected line graph is hamiltonian is valid.