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OP-10080Status unassessedCombinatorics

Written on the Wall II - Conjecture 198a

Written on the Wall II - Conjecture 198a

MSC 05 · Combinatorics

Inferred from the community field; finer classification has not been reviewed.

Specialist source · status unassessedAs of 2026-09-10Source & evidence ↓
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Problem statement

conjecture198a

WOWII Conjecture 198a

For a simple connected graph G, if b(G) ≤ 2 + ecc_avg(G), then G has a Hamiltonian path. Here b(G) is the number of vertices in a largest induced bipartite subgraph, and ecc_avg(G) is the average eccentricity of G. A Hamiltonian path is a walk visiting every vertex exactly once.

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Formal statement

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/-
Copyright 2025 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

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

/-!
# Written on the Wall II - Conjecture 198a

*Reference:*
[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)
-/

namespace WrittenOnTheWallII.GraphConjecture198a

open SimpleGraph

variable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]

/--
WOWII [Conjecture 198a](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)

For a simple connected graph `G`, if `b(G) ≤ 2 + ecc_avg(G)`, then `G` has a Hamiltonian path.
Here `b(G)` is the number of vertices in a largest induced bipartite subgraph, and
`ecc_avg(G)` is the average eccentricity of `G`.
A Hamiltonian path is a walk visiting every vertex exactly once.
-/
@[category research open, AMS 5]
theorem conjecture198a (G : SimpleGraph α) (h : G.Connected)
    (hb : b G ≤ 2 + averageEccentricity G) :
    ∃ a b : α, ∃ p : G.Walk a b, p.IsHamiltonian := by
  sorry

-- Sanity checks

/-- The `averageEccentricity` invariant is nonneg for any graph. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ averageEccentricity G := by
  unfold averageEccentricity
  positivity

/-- The invariant `b G` is nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _

end WrittenOnTheWallII.GraphConjecture198a

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Background

Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.

Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.

Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.

References
  1. [1]Formal Conjectures — Written on the Wall II - Conjecture 198a link
  2. [2]Original reference cited in the formalization link

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