The Bollobás–Eldridge–Catlin conjecture on graph packing (1978)

The Bollobás–Eldridge–Catlin conjecture (1978). If G_1 and G_2 are graphs on the same n vertices with maximum degrees Δ_1, Δ_2 satisfying (Δ_1 + 1)(Δ_2 + 1) ≤ n + 1, then G_1 and G_2 pack.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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@misc{cairn-paper-bollobas-eldridge-catlin-conjecture,
  title        = {The Bollobás–Eldridge–Catlin conjecture on graph packing (1978)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-bollobas-eldridge-catlin-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

The Bollobás–Eldridge–Catlin conjecture (1978).

If $G_{1}$ and $G_{2}$ are graphs on the same $n$ vertices with maximum degrees $Δ_{1}$ , $Δ_{2}$ satisfying $(Δ_{1} + 1) (Δ_{2} + 1) \leq n + 1$ , then $G_{1}$ and $G_{2}$ pack.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.BollobasEldridgeCatlinConjecture.

theorem bollobas_eldridge_catlin_conjecture :
    ∀ {V : Type} [Fintype V] [DecidableEq V] (G₁ G₂ : SimpleGraph V)
      [DecidableRel G₁.Adj] [DecidableRel G₂.Adj],
      (G₁.maxDegree + 1) * (G₂.maxDegree + 1) ≤ Fintype.card V + 1 → Packs G₁ G₂

What counts as progress

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References

[BE78] Bollobás, B. and Eldridge, S. E. (1978). "Packings of graphs and applications to computational complexity." J. Combin. Theory Ser. B 25, pp. 105--124.
[Ca74] Catlin, P. A. (1974). "Subgraphs of graphs, I." Discrete Math. 10, pp. 225--233.
[SS78] Sauer, N. and Spencer, J. (1978). "Edge disjoint placement of graphs." J. Combin. Theory Ser. B 25, pp. 295--302.
[AF93] Aigner, M. and Brandt, S. (1993). "Embedding arbitrary graphs of maximum degree two." J. London Math. Soc. 48, pp. 39--51.
[CSS03] Csaba, B., Shokoufandeh, A. and Szemerédi, E. (2003). "Proof of a conjecture of Bollobás and Eldridge for graphs of maximum degree three." Combinatorica 23, pp. 35--72.

Source and licence

Imported from Formal Conjectures (research papers), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.