A new approach towards a conjecture on intersecting three longest paths

摘要

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 much work has been done on this topic. One important open question in this area is to ask whether any three longest paths contain 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.