Nash Williams Conjecture and the Dominating Cycle Conjecture

TL;DR

This paper explores the relationship between the Nash Williams conjecture and the Dominating Cycle Conjecture, showing that a modified version of the former is equivalent to the latter, thereby linking two important conjectures in graph theory.

Contribution

It establishes an equivalence between a modified Nash Williams conjecture and the Dominating Cycle Conjecture, providing new insights into their interrelation.

Findings

Modified Nash Williams conjecture is equivalent to the Dominating Cycle Conjecture

Disproof of original Nash Williams conjecture related to Hamiltonian cycles in 4-regular graphs

New theoretical connection between two longstanding conjectures in graph theory

Abstract

The disproved Nash Williams conjecture states that every 4-regular 4-connected graph has a hamiltonian cycle. We show that a modification of this conjecture is equivalent to the Dominating Cycle Conjecture.