feat: Add K3 verification for Conjecture 19 #1511
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Prove that K3 (complete graph on 3 vertices) satisfies Conjecture 19:
- K3_b: b(K3) = 2 (largest induced bipartite subgraph size)
- K3_ecc: ecc(K3, {v}) = 1 for all vertices
- K3_indepNeighbors: indepNeighbors(K3, v) = 1 for all vertices
- K3_not_counterexample: Main theorem showing K3 satisfies the conjecture
Includes helper lemma indepNum_completeGraph proving the independence
number of a complete graph is 1.
mo271
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Jan 7, 2026
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looks good.
Just to check: is this a legit counter example to Conjecture 19 on
https://web.archive.org/web/20250908021324/http://cms.dt.uh.edu/faculty/delavinae/research/wowII/
or are there implicit, i.e. not captured by our formalisation, assumptions that exclude this (very small!) counterexample?
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| rw [SimpleGraph.b, SimpleGraph.largestInducedBipartiteSubgraphSize] | ||
| norm_cast | ||
| apply le_antisymm | ||
| · -- b K3 <= 2 | ||
| apply csSup_le | ||
| · use 0 | ||
| use ∅ | ||
| simp | ||
| dsimp [SimpleGraph.IsBipartite] | ||
| apply SimpleGraph.colorable_of_isEmpty | ||
| · rintro n ⟨s, hs⟩ | ||
| by_contra h_gt | ||
| simp at h_gt | ||
| have h_card : s.card = 3 := by | ||
| have : s.card ≤ 3 := Finset.card_le_univ s | ||
| linarith | ||
| have h_univ : s = Finset.univ := Finset.eq_univ_of_card s h_card | ||
| rw [h_univ] at hs | ||
| have h_bip : K3.IsBipartite := by | ||
| have h_iso : K3.induce (Set.univ : Set (Fin 3)) ≃g K3 := SimpleGraph.induceUnivIso K3 | ||
| rw [← Finset.coe_univ] at h_iso | ||
| exact hs.1.of_embedding h_iso.symm.toEmbedding | ||
| have h_not_bip : ¬ K3.IsBipartite := by | ||
| dsimp [SimpleGraph.IsBipartite] | ||
| rw [← SimpleGraph.chromaticNumber_le_iff_colorable] | ||
| simp [K3, completeGraph, SimpleGraph.chromaticNumber_top] | ||
| norm_num | ||
| contradiction | ||
| · -- b K3 >= 2 | ||
| apply le_csSup | ||
| · use 3 | ||
| rintro n ⟨s, hs⟩ | ||
| rw [← hs.2] | ||
| exact Finset.card_le_univ s | ||
| · use {0, 1} | ||
| constructor | ||
| · -- K3.induce {0, 1} is bipartite (it's K2) | ||
| dsimp [SimpleGraph.IsBipartite] | ||
| fconstructor | ||
| refine SimpleGraph.Coloring.mk (fun x => if (x : Fin 3) = 0 then 0 else 1) ?_ | ||
| intro u v huv | ||
| have huv_ne : u ≠ v := SimpleGraph.Adj.ne huv | ||
| have u_mem : (u : Fin 3) ∈ ({0, 1} : Finset (Fin 3)) := u.property | ||
| have v_mem : (v : Fin 3) ∈ ({0, 1} : Finset (Fin 3)) := v.property | ||
| simp only [Finset.mem_insert, Finset.mem_singleton] at u_mem v_mem | ||
| rcases u_mem with u_eq_0 | u_eq_1 <;> rcases v_mem with v_eq_0 | v_eq_1 | ||
| · -- u = 0, v = 0 | ||
| exfalso; apply huv_ne; ext; simp only [u_eq_0, v_eq_0] | ||
| · -- u = 0, v = 1 | ||
| simp only [u_eq_0, v_eq_1, ↓reduceIte] | ||
| exact zero_ne_one | ||
| · -- u = 1, v = 0 | ||
| simp only [u_eq_1, v_eq_0, ↓reduceIte, one_ne_zero] | ||
| exact zero_ne_one.symm | ||
| · -- u = 1, v = 1 | ||
| exfalso; apply huv_ne; ext; simp only [u_eq_1, v_eq_1] | ||
| · simp |
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Suggested change
| rw [SimpleGraph.b, SimpleGraph.largestInducedBipartiteSubgraphSize] | |
| norm_cast | |
| apply le_antisymm | |
| · -- b K3 <= 2 | |
| apply csSup_le | |
| · use 0 | |
| use ∅ | |
| simp | |
| dsimp [SimpleGraph.IsBipartite] | |
| apply SimpleGraph.colorable_of_isEmpty | |
| · rintro n ⟨s, hs⟩ | |
| by_contra h_gt | |
| simp at h_gt | |
| have h_card : s.card = 3 := by | |
| have : s.card ≤ 3 := Finset.card_le_univ s | |
| linarith | |
| have h_univ : s = Finset.univ := Finset.eq_univ_of_card s h_card | |
| rw [h_univ] at hs | |
| have h_bip : K3.IsBipartite := by | |
| have h_iso : K3.induce (Set.univ : Set (Fin 3)) ≃g K3 := SimpleGraph.induceUnivIso K3 | |
| rw [← Finset.coe_univ] at h_iso | |
| exact hs.1.of_embedding h_iso.symm.toEmbedding | |
| have h_not_bip : ¬ K3.IsBipartite := by | |
| dsimp [SimpleGraph.IsBipartite] | |
| rw [← SimpleGraph.chromaticNumber_le_iff_colorable] | |
| simp [K3, completeGraph, SimpleGraph.chromaticNumber_top] | |
| norm_num | |
| contradiction | |
| · -- b K3 >= 2 | |
| apply le_csSup | |
| · use 3 | |
| rintro n ⟨s, hs⟩ | |
| rw [← hs.2] | |
| exact Finset.card_le_univ s | |
| · use {0, 1} | |
| constructor | |
| · -- K3.induce {0, 1} is bipartite (it's K2) | |
| dsimp [SimpleGraph.IsBipartite] | |
| fconstructor | |
| refine SimpleGraph.Coloring.mk (fun x => if (x : Fin 3) = 0 then 0 else 1) ?_ | |
| intro u v huv | |
| have huv_ne : u ≠ v := SimpleGraph.Adj.ne huv | |
| have u_mem : (u : Fin 3) ∈ ({0, 1} : Finset (Fin 3)) := u.property | |
| have v_mem : (v : Fin 3) ∈ ({0, 1} : Finset (Fin 3)) := v.property | |
| simp only [Finset.mem_insert, Finset.mem_singleton] at u_mem v_mem | |
| rcases u_mem with u_eq_0 | u_eq_1 <;> rcases v_mem with v_eq_0 | v_eq_1 | |
| · -- u = 0, v = 0 | |
| exfalso; apply huv_ne; ext; simp only [u_eq_0, v_eq_0] | |
| · -- u = 0, v = 1 | |
| simp only [u_eq_0, v_eq_1, ↓reduceIte] | |
| exact zero_ne_one | |
| · -- u = 1, v = 0 | |
| simp only [u_eq_1, v_eq_0, ↓reduceIte, one_ne_zero] | |
| exact zero_ne_one.symm | |
| · -- u = 1, v = 1 | |
| exfalso; apply huv_ne; ext; simp only [u_eq_1, v_eq_1] | |
| · simp | |
| simp only [GraphConjecture19.K3, b] | |
| norm_cast | |
| refine IsGreatest.csSup_eq ?_ | |
| constructor | |
| · simp [SimpleGraph.IsBipartite] | |
| use {0,1} | |
| constructor | |
| · use fun i => ite (i.1 = 0) 0 1 | |
| simp | |
| decide | |
| · decide | |
| · simp only [upperBounds, Finset.coe_sort_coe, completeGraph_eq_top, Set.mem_setOf_eq, forall_exists_index, | |
| and_imp, forall_apply_eq_imp_iff₂] | |
| intro a ha | |
| cases ha | |
| cases ‹_› | |
| decide+revert |
I think this could even be simplified further.
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I suggest to put this directly in the file
FormalConjectures/WrittenOnTheWallII/GraphConjecture19.lean,
Perhaps using the answer(False) \iff mechanism: this way we avoid copying the statement.
And let's optimize the proofs a bit...
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Just to clarify, this is not a counterexample. Millions of graphs have already been computationally tested against this conjecture (which was generated by an automated system, see e.g. https://www.sciencedirect.com/science/article/pii/S0004370215001575 for more details) without contradiction. The goal of this test is simply to establish a very basic baseline that we can expand to include more graphs later.
Paul-Lez
reviewed
Jan 8, 2026
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| import Mathlib.Combinatorics.SimpleGraph.Coloring | ||
| import Mathlib.Combinatorics.SimpleGraph.Bipartite |
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I think this should work
Suggested change
| import Mathlib.Combinatorics.SimpleGraph.Coloring | |
| import Mathlib.Combinatorics.SimpleGraph.Bipartite |
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Summary
Prove that K3 (complete graph on 3 vertices) satisfies Conjecture 19.
Changes
b(K3) = 2(largest induced bipartite subgraph size)ecc(K3, {v}) = 1for all verticesindepNeighbors(K3, v) = 1for all verticesIncludes helper lemma
indepNum_completeGraphproving the independence number of a complete graph is 1.Testing
lake buildsorryplaceholders remain in the file