Improvements on Hippchen's Conjecture

TL;DR

This paper advances the understanding of Hippchen's Conjecture by proving it holds for additional values of k, specifically k=5 and k≥(n+2)/5, thereby resolving two of Gutiérrez's conjectures.

Contribution

The authors extend the validity of Hippchen's Conjecture to new ranges of k, specifically for k=5 and k≥(n+2)/5, improving previous results.

Findings

Proved the conjecture for k=5.

Proved the conjecture for k≥(n+2)/5.

Resolved two conjectures of Gutiérrez affirmatively.

Abstract

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. Guti\'errez 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 Guti\'errez in the affirmative.