On two conjectures about the intersection of longest paths and cycles

Abstract

A conjecture attributed to Smith states that every pair of longest cycles in a $k$-connected graph intersect each other in at least $k$ vertices. In this paper, we show that every pair of longest cycles in a~$k$-connected graph on $n$ vertices intersect each other 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 $k\le 6$ or $k\ge (n+9)/7$.