Hadwiger's conjecture implies a conjecture of F\"uredi-Gy\'arf\'as-Simonyi

TL;DR

This paper shows that if Hadwiger's conjecture is true, then a related conjecture by F"uredi-Gyárfás-Simonyi about connected matchings in graphs with independence number 2 also holds, linking two major open problems in graph theory.

Contribution

The paper establishes that F"uredi-Gyárfás-Simonyi's conjecture is a consequence of Hadwiger's conjecture, highlighting a significant connection between these two conjectures in graph theory.

Findings

F"uredi-Gyárfás-Simonyi's conjecture follows from Hadwiger's conjecture.

If F"uredi-Gyárfás-Simonyi's conjecture is false, then Hadwiger's conjecture is also false.

The work links the validity of two major open problems in graph theory.

Abstract

One of the most important open problems in the field of graph colouring or even graph theory is the conjecture of Hadwiger. This conjecture was the inspiration for many mathematical works, one of them being the work of F\"uredi, Gy\'arf\'as and Simonyi in which they "risked" to conjecture the precise bound for a graph with independence number $2$ to contain a certain connected matching. We prove that their conjecture would be a corollary of Hadwiger's conjecture or equivalently if their risky conjecture would be false, then Hadwiger's conjecture would be false as well.