Erdős Problem #742 — open problem in graph theory

Erdős Problem #742

Murty-Simon Conjecture Let G be a graph on n vertices with diameter 2 such that deleting any edge increases the diameter. Is it true that G has at most ⌊ n^2 / 4 ⌋ edges? Equality is conjectured to hold for the complete balanced bipartite graph K_⌈ n/2 ⌉, ⌊ n/2 ⌋.

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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Cite

@misc{cairn-erdos-742,
  title        = {Erdős Problem #742},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-742}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Murty-Simon Conjecture

Let $G$ be a graph on $n$ vertices with diameter $2$ such that deleting any edge increases the diameter. Is it true that $G$ has at most $⌊ n^{2} /4 ⌋$ edges? Equality is conjectured to hold for the complete balanced bipartite graph $K_{⌈ n /2 ⌉, ⌊ n /2 ⌋}$ .

The conjecture is resolved up to a finite check: Fan [Fa87] verified it for $n \leq 24$ and $n = 26$ , and Füredi [Fü92] proved it for all sufficiently large $n$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«742». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_742 :
    answer(sorry) ↔ ∀ (V : Type*) [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj], IsDiameter2Critical G →
    G.edgeFinset.card ≤ (Fintype.card V) ^ 2 / 4

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/742. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/742
[Pl75] Plesník, Ján, Critical graphs of given diameter. Acta Fac. Rerum Natur. Univ. Comenian. Math. 30 (1975), 71-93.
[CaHa79] Caccetta, L. and Häggkvist, R., On diameter critical graphs. Discrete Math. 28 (1979), 223-229.
[Fa87] Fan, Genghua, On diameter 2-critical graphs. Discrete Math. 67 (1987), 235-240.
[Fü92] Füredi, Zoltán, The maximum number of edges in a minimal graph of diameter 2. J. Graph Theory 16 (1992), 81-98.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.