# Equivalent formulation of Thomassen's conjecture using Tutte paths in claw-free graphs

## TL;DR

This paper explores an equivalent formulation of Thomassen's conjecture, linking Hamilton cycles in line graphs to Tutte paths in claw-free graphs, and extends the concept to connect any two vertices with a maximal Tutte path.

## Contribution

It establishes a new equivalent formulation of Thomassen's conjecture involving Tutte paths in claw-free graphs and extends the framework to connect any two vertices with such paths.

## Key findings

- Proves the equivalence between Thomassen's and Jackson's conjectures.
- Extends the formulation to connect any two vertices with a Tutte path.
- Provides new insights into Hamiltonicity in claw-free graphs.

## Abstract

We continue studying Thomassen's conjecture (every 4-connected line graph has a Hamilton cycle) in the direction of a recently shown equivalence with Jackson's conjecture (every 2-connected claw-free graph has a Tutte cycle), and we extend the equivalent formulation as follows: In each connected claw-free graph, every two vertices are connected by a maximal path which is a Tutte path.