FormalConjecturesForMathlib.Combinatorics.SimpleGraph.Hypercube

/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
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-/
module


public import Mathlib.Combinatorics.SimpleGraph.Basic
public import Mathlib.Data.Fintype.Pi

@[expose] public section

The hypercube graph

SimpleGraph.hypercube n is the graph Qₙ on the n-bit vectors, with two vectors adjacent when they differ in exactly one coordinate. It has 2 ^ n vertices and n * 2 ^ (n - 1) edges.

namespace SimpleGraphopen scoped Finset

hypercube n is the n-dimensional hypercube graph Qₙ: the vertices are the n-bit vectors, and two vertices are adjacent when they differ in exactly one coordinate.

def hypercube (n : ℕ) : SimpleGraph (Fin n → Bool) where
  Adj u v := #{i | u i ≠ v i} = 1
  symm.symm _ _ := byn:ℕx✝¹:Fin n → Boolx✝:Fin n → Bool⊢ #{i | x✝¹ i ≠ x✝ i} = 1 → #{i | x✝ i ≠ x✝¹ i} = 1 simp [eq_comm]All goals completed! 🐙
  loopless.irrefl _ := byn:ℕx✝:Fin n → Bool⊢ ¬#{i | x✝ i ≠ x✝ i} = 1 simpAll goals completed! 🐙

@[simp]
theorem hypercube_adj {n : ℕ} {u v : Fin n → Bool} :
    (hypercube n).Adj u v ↔ #{i | u i ≠ v i} = 1 := Iff.rfl

end SimpleGraph