Subgraphs of large connectivity and chromatic number

Abstract

Resolving a problem raised by Norin in 2020, we show that for each k ∈ N $k \in \mathbb {N}$ , the minimal f ( k ) ∈ N $f(k) \in \mathbb {N}$ with the property that every graph G $G$ with chromatic number at least f ( k ) + 1 $f(k)+1$ contains a subgraph H $H$ with both connectivity and chromatic number at least k $k$ satisfies f ( k ) ⩽ 7 k $f(k) \leqslant 7k$ . This result is best-possible up to multiplicative constants, and sharpens earlier results of Alon–Kleitman–Thomassen–Saks–Seymour from 1987 showing that f ( k ) = O ( k 3 ) $f(k) = O(k^3)$ , and of Chudnovsky–Penev–Scott–Trotignon from 2013 showing that f ( k ) = O ( k 2 ) $f(k) = O(k^2)$ . Our methods are robust enough to handle list colouring as well: we additionally show that for each k ∈ N $k \in \mathbb {N}$ , the minimal f ℓ ( k ) ∈ N $f_\ell (k) \in \mathbb {N}$ with the property that every graph G $G$ with list chromatic number at least f ℓ ( k ) + 1 $f_\ell (k)+1$ contains a subgraph H $H$ with both connectivity and list chromatic number at least k $k$ is well-defined and satisfies f ℓ ( k ) ⩽ 4 k $f_\ell (k) \leqslant 4k$ . This result is again best-possible up to multiplicative constants; here, unlike with f ( · ) $f(\cdot )$ , even the existence of f ℓ ( · ) $f_\ell (\cdot )$ appears to have been previously unknown.