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

TL;DR

This paper reframes Thomassen's conjecture through Tutte-path formulations in claw-free graphs, aiming to unify several equivalent perspectives on Hamiltonicity. It proposes a new maximal $(a,b)$-path Tutte-path formulation (Conjecture 1) for every pair of vertices $a,b$ in a connected claw-free graph and proves its equivalence to the classic conjecture via a Tutte-closure operation. The authors develop a structural toolkit by reviewing Tutte paths and closures, and then connect these ideas to line graphs of multigraphs and hypergraphs, leveraging Bermond–Meyer's forbidden-subgraph characterization and the rank-$3$ hypergraph framework. This approach yields a robust bridge between Thomassen's conjecture and line-graph/hypergraph structure, offering a pathway to prove the conjecture through Tutte-closure reductions and known line-graph results. The work thus advances the understanding of Hamiltonicity in claw-free and line-graph contexts with potential implications for related conjectures such as Jackson's and Li et al.'s formulations.

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.