Smith’s conjecture on intersections of longest cycles

Prove that for every integer k≥2, any two longest cycles in a k-connected graph share at least k vertices.

Background

Smith’s conjecture is the central problem addressed by the paper. The paper establishes only a linear-factor bound, showing that two longest cycles in a k-connected graph share at least k/600 vertices, so the conjectured sharp constant remains unresolved.

The conjectured bound is sharp, as demonstrated by complete bipartite graphs K_{k,n−k} with n≥3k. Before this work, the conjecture was known only for k≤8 and for certain sufficiently dense graphs.