Publications

On the extended Clark-Wormold Hamiltonian-like index problem.

Faculty Author(s): Lai, Hong-Jian
Student Author(s): -
Department: MATH
Publication: Discrete Mathematics
Year: 2022
Abstract: Summary: ``For a hamiltonian property PP, Clark and Wormold introduced the problem of investigating the value P(a,b)=max{min{n:Ln(G)P(a,b)=max{min{n:Ln(G) has property P}P}: κ′(G)≥aκ′(G)≥a and δ(G)≥b}δ(G)≥b}, and proposed a few problems to determine P(a,b)P(a,b) with b≥a≥4b≥a≥4 when PP 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)ess′(G) denote the essential edge-connectivity of a graph GG, and define P′(a,b)=max{min{n:Ln(G)P′(a,b)=max{min{n:Ln(G) has property P}P}: ess′(G)≥aess′(G)≥a and δ(G)≥b}δ(G)≥b}. We investigate the values of P′(a,b)P′(a,b) when PP is one of these hamiltonian properties. In particular, we show that for any values of b≥1b≥1, P′(4,b)≤2P′(4,b)≤2 and P′(4,b)=1P′(4,b)=1 if and only if Thomassen's conjecture that every 4-connected line graph is hamiltonian is valid.''
Link: On the extended Clark-Wormold Hamiltonian-like index problem.