Wagner 1964: no $K_{t+1}$ minor $\Rightarrow$ $2^t$ -colourable

Proved

HadwigerConj.wagner_colorable_two_pow

by Lucas · Sep 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

graph-coloringgraph-minorsgraph-theory

For every integer $t\ge 0$ , every finite graph $G$ with no $K_{t+1}$ minor is $2^t$ -colourable:

K_{t+1}\not\preceq G\ \Longrightarrow\ \chi(G)\le 2^{t}.

This is the first general bound on the chromatic number of graphs excluding a complete minor.

Preamble

import Mathlib
import Definitions.Def_HadwigerConj_Defs

Formal statement

namespace HadwigerConj
theorem wagner_colorable_two_pow (t : ℕ) {V : Type} [Finite V] (G : SimpleGraph V)
    (hG : ¬ HasCompleteMinor G (t + 1)) : G.Colorable (2 ^ t) := by sorry
end HadwigerConj

Source

P. Seymour, "Hadwiger's conjecture" (survey), in: Open Problems in Mathematics, Springer, 2016 (uploaded PDF `paper.pdf`), Section 3, first paragraph (p. 4: “As Wagner proved in 1964, all graphs with no K_{t+1} minor are 2^t-colourable”)

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

Note — non-blind read-back. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements, with full knowledge of the intended meaning. It is not independent or blind testimony, and no reviewer should treat it as such; an independent audit is still needed.

For every natural number $t$ , every finite type $V$ and every simple graph $G$ on $V$ : if $G$ has no $K_{t+1}$ minor then $G$ is properly colourable with $2^t$ colours. Here, for graphs $H$ on a vertex type $W$ and $G$ on a vertex type $V$ , “ $H$ is a minor of $G$ ” means: there are sets $B_w\subseteq V$ ( $w\in W$ ) such that (i) each $B_w$ induces a connected subgraph of $G$ (in particular $B_w\neq\emptyset$ ), (ii) $B_{w_1}\cap B_{w_2}=\emptyset$ whenever $w_1\neq w_2$ , and (iii) whenever $w_1w_2$ is an edge of $H$ , some vertex of $B_{w_1}$ is adjacent in $G$ to some vertex of $B_{w_2}$ . “ $G$ has a $K_t$ minor” means the complete graph on the $t$ -element set $\{0,\dots,t-1\}$ is a minor of $G$ in this sense (so: $t$ pairwise disjoint connected nonempty vertex sets, every two joined by an edge of $G$ ; for $t=0$ this holds for every graph). “ $k$ -colourable” means there is a map from $V$ to a $k$ -element set of colours giving adjacent vertices different colours (for $k=0$ this forces $V=\emptyset$ ). All graphs are simple (no loops, no multiple edges).

Human review

Endorsed by Shuze Chen · Oct 1, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 1, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Wagner 1964: no Kt+1K_{t+1}Kt+1​ minor ⇒\Rightarrow⇒2t2^t2t-colourable

Wagner 1964: no $K_{t+1}$ minor $\Rightarrow$ $2^t$ -colourable