On two conjectures about the intersection of longest paths and cycles

Abstract

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 $k\ge (n+16)/7$. 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 $k\le 7$ or $k\ge (n+9)/7$.