论文标题
河本的猜想的改进
Improvements on Hippchen's Conjecture
论文作者
论文摘要
令$ g $为$ n $顶点上的$ k $连接图。希普的猜想指出，$ g $共享的两条最长的路径至少至少$ k $顶点。 Gutiérrez最近证明了当$ k \ leq 4 $或$ k \ geq \ frac {n-2} {3} $时证明了猜想。我们在这两个结果上有所改善；也就是说，我们表明，$ g $共享的两条最长路径至少$ k $顶点当$ k = 5 $或$ k \ geq \ frac {n+2} {5} $。这完全解决了古蒂尔雷斯的两个猜想。
Let $G$ be a $k$-connected graph on $n$ vertices. Hippchen's Conjecture states that two longest paths in $G$ share at least $k$ vertices. Gutiérrez recently proved the conjecture when $k\leq 4$ or $k\geq \frac{n-2}{3}$. We improve upon both results; namely, we show that two longest paths in $G$ share at least $k$ vertices when $k=5$ or $k\geq \frac{n+2}{5}$. This completely resolves two conjectures of Gutiérrez in the affirmative.
