Hadwiger Number Challenge

1. About the Problem

The Hadwiger number h(G) of a graph G is the largest integer t such that the complete graph K_t is a minor of G. It is among the most natural measures of graph complexity, yet exact values remain unknown for surprisingly small cases.

Hadwiger's Conjecture (1943) states that every graph with no K_t minor is (t-1)-colorable. This is one of the most famous open problems in graph theory.

Status in Mathlib4

The Hadwiger number and Hadwiger's conjecture are not currently formalized in Mathlib4. Related concepts available: SimpleGraph, completeGraph, and basic graph coloring in Mathlib.Combinatorics.SimpleGraph.

2. The Challenge

The file HadwigerChallenge/Basic.lean contains two sorries:

Define hadwigerNumber — the largest t such that K_t is a minor of G.
Prove hadwiger_challenge — for parameter k = 2, show that every graph with Hadwiger number at most k is (k+1)-colorable.

In mathematical terms: h(G) ≤ 2 → G is 3-colorable. This is a special case of Hadwiger's conjecture.

Code authorship is open — human-written, AI-generated, or mixed.

4. Submit

The challenge file is preloaded below. Edit it directly, or paste your own code.

HadwigerChallenge/Basic.lean

/-
# Hadwiger Number Challenge

A Lean 4 formalization challenge. Fill in the sorries below.

## The Problem

The Hadwiger number `h(G)` of a graph `G` is the largest integer `t`
such that the complete graph `K_t` is a minor of `G`.

Hadwiger's Conjecture (1943): every graph with no `K_t` minor is `(t-1)`-colorable.

This file challenges you to prove a special case for `k = 2`.

## What you need to do

1. Replace `sorry` in `hadwigerNumber` with a proper definition.
2. Replace `sorry` in `hadwiger_challenge` with a complete proof.

No other `sorry` tactics are permitted in your submission.
-/

import Mathlib

-- The challenge parameter
-- We fix k = 2: prove the statement for graphs with Hadwiger number at most 2.
def k : ℕ := 2

-- Define the Hadwiger number of a graph.
-- This is the largest t such that K_t is a minor of G.
def hadwigerNumber {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) : ℕ :=
  sorry

-- Challenge: Prove that every graph with Hadwiger number at most k is (k+1)-colorable.
--
-- For k = 2, this reads:
--   If the largest complete minor of G has size at most 2,
--   then G is 3-colorable.
--
-- Equivalently: a graph with no K_3 minor is 3-colorable.
-- (This is a special case of Hadwiger's conjecture.)
theorem hadwiger_challenge {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) :
    hadwigerNumber G ≤ k → G.Colorable (k + 1) := by
  sorry

Click "Check for Sorry" to scan.

Note: "Check for Sorry" is client-side only. For full compilation, use "Open in Playground" which sends your code to live.lean-lang.org.

#	Participant	Graph Class	Bound	Lines	Date
1	alice_prover	K₅-minor-free	h(G) ≤ 4	127	2026-05-10
2	graph_formalist	Planar graphs	h(G) ≤ 4	203	2026-05-08
3	lean_bot_7	Forests	h(G) ≤ 2	89	2026-05-12

Hadwiger Number Challenge

1. About the Problem

2. The Challenge

3. Rules

4. Submit

5. Leaderboard