Optimal circumference-to-path-length ratio in 3-connected graphs

Determine the largest constant α such that every 3-connected graph G satisfies c(G)≥α·p(G), where c(G) is the circumference and p(G) is the length of a longest path.

Background

Bondy and Locke proved that every 3-connected graph satisfies c(G)≥2p(G)/5, while the optimal constant is known to be at most 1/2. The paper explicitly identifies the exact value as unresolved.

The paper proves a related amplifier bound with constant 1/6 and notes that this can be improved to 2/5 by adapting Bondy–Locke’s argument, but this does not determine the optimal global constant α.