On two conjectures about the intersection of longest paths and cycles

A conjecture attributed to Smith states that every two longest cycles in a k-connected graph intersect in at least k vertices. In this paper, we show that every two longest cycles in a k-connected graph on n vertices intersect in at least min⁡{n,8k−n−16} vertices, which confirms Smith's conjecture when k≥(n+16)/7. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either k≤7 or k≥(n+9)/7.