A New Approach Towards a Conjecture on Intersecting Three Longest Paths

TL;DR

This paper introduces a new approach based on distances among longest paths in connected graphs, making significant progress towards proving the conjecture that any three longest paths share a common vertex.

Contribution

It proposes a novel method focusing on distances among longest paths, advancing the understanding of the conjecture about intersecting three longest paths.

Findings

Progress towards the conjecture on three longest paths

New approach based on distances among longest paths

Substantial theoretical advancement

Abstract

In 1966, T. Gallai asked whether every connected graph has a vertex that appears in all longest paths. Since then this question has attracted much attention and many work has been done in this topic. One important open question in this area is to ask whether any three longest paths contains a common vertex in a connected graph. It was conjectured that the answer to this question is positive. In this paper, we propose a new approach in view of distances among longest paths in a connected graph, and give a substantial progress towards the conjecture along the idea.