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

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.