Bounding the distance among longest paths in a connected graph

TL;DR

This paper investigates bounds on the distance among the longest paths in a connected graph, extending previous results to cases involving four or more longest paths and providing general bounds for any number k.

Contribution

It introduces new upper bounds on the distance among k longest paths in a connected graph, generalizing prior work focused on fewer paths.

Findings

Established an upper bound for the distance among 4 longest paths.

Extended the bounds to k longest paths for any k.

Provided theoretical insights into the structure of longest paths in graphs.

Abstract

It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k>=7, Skupien in [7] obtained a connected graph in which some k longest paths have no common vertex, but every k-1 longest paths have a common vertex. It is not known whether every 3 longest paths in a connected graph have a common vertex and similarly for 4, 5, and 6 longest path. In [5] the authors give an upper bound on distance among 3 longest paths in a connected graph. In this paper we give a similar upper bound on distance between 4 longest paths and also for k longest paths, in general.