Proof of Thomassen's Conjecture on Highly connected subgraphs with large chromatic number

TL;DR

This paper proves Thomassen's conjecture by establishing an upper bound on the function g(k,m), showing that large chromatic number graphs contain highly connected subgraphs with large chromatic number.

Contribution

It provides a new proof of Thomassen's conjecture using Hall-feasibility, improving understanding of highly connected subgraphs in chromatic graphs.

Findings

Proved that g(k,m) ≤ max(m+2k-2, 3k+1) for all k ≥ 1, m ≥ 2.

Confirmed Thomassen's conjecture that g(k,k+1) ≤ 3k+1.

Introduced a Hall-feasibility argument as a key proof technique.

Abstract

For integers $k\ge 1$ and $m\ge 2$, let $g(k,m)$ be the least integer $n\ge 1$ such that every graph with chromatic number at least $n$ contains a $(k+1)$-connected subgraph with chromatic number at least $m$. We prove that \[ g(k,m)\le \max(m+2k-2,\,3k+1) \] for all $k\ge 1$ and $m\ge 2$, establishing the 1983 conjecture of Thomassen that $g(k,k+1)\le 3k+1$. The key new ingredient is a Hall-feasibility argument replacing the final numerical step in the proof of Nguyen.