Hippchen’s conjecture on intersections of longest paths

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

Background

Hippchen’s conjecture is presented as the longest-path analogue of Smith’s conjecture. The paper proves only the weaker bound that two longest paths share at least floor(k/600) vertices.

The authors note that the standard reduction obtained by adding a universal vertex transfers Smith-type results for cycles to longest paths, but the exact conjectured lower bound k remains unresolved in the paper.

References

For every $k\geq 2$, any two longest paths in a $k$-connected graph have at least $k$ common vertices.

— Longest cycles intersect linearly in highly connected graphs  (2609.20724 - Ma et al., 17 Sep 2026) in Section 6.1, “A linear bound for Hippchen’s conjecture”