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 $k\geq (n+16)/7$. 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 $k \leq 6$ or $k \geq (n+9)/7$.

翻译：史密斯提出的猜想指出，在$k$-连通图中，每对最长循环至少相交于$k$个顶点。本文证明，在拥有$n$个顶点的$k$-连通图中，每对最长循环至少相交于$\min\{n,8k-n-16\}$个顶点，从而当$k\geq (n+16)/7$时确认了史密斯猜想的正确性。希普钦提出了关于路径而非循环的类似猜想。通过简单归约，我们将两个猜想关联起来，证明当$k \leq 6$或$k \geq (n+9)/7$时希普钦猜想成立。