Optimal amplifier constant under three internally disjoint paths

Determine the optimal constant α such that, whenever a multigraph H contains three internally disjoint (a,b)-paths, every (a,b)-path P in H is contained in a cycle C through a and b with |E(C)∩E(P)|≥α|P|.

Background

Theorem 3 proves that the constant 1/6 is always attainable under the stated three-path condition. The authors explicitly state that this constant is not best possible and that their argument can be improved to 2/5 with greater technical complexity.

Determining the optimal amplifier constant is presented as a possible route toward understanding the broader Bondy–Locke circumference-to-path-length question.