Any Three Longest Paths In A Connected Graph Has A Common Vertex

TL;DR

This paper proves that in any connected graph, any three longest paths share at least one common vertex, confirming a longstanding question from 1995.

Contribution

It establishes that three longest paths in a connected graph always have a common vertex, resolving a question posed over two decades ago.

Findings

Any three longest paths in a connected graph have a common vertex.

The result confirms a 1995 conjecture in graph theory.

Provides a definitive answer to a long-standing open problem.

Abstract

A question was raised in 1995 at the British Combinatorial Conference: Do any three longest paths in a connected graph have a vertex in common? In this paper, it is shown that the answer to that question is yes.