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Erdős #1155 · graph theory

Construct a random graph on nn vertices in the following way: begin with the complete graph KnK_n. At each stage, choose uniformly a random triangle in the graph and delete all the edges of this triangle. Repeat until t…

MSC 05C · Graph theory

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Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Construct a random graph on nn vertices in the following way: begin with the complete graph KnK_n. At each stage, choose uniformly a random triangle in the graph and delete all the edges of this triangle. Repeat until the graph is triangle-free.

Describe the typical parameters and structure of such a graph. In particular, if f(n)f(n) is the number of edges remaining, then is it true that

Ef(n)≍n3/2\mathbb{E}f(n)\asymp n^{3/2}

and that f(n)≪n3/2f(n) \ll n^{3/2} almost surely?

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Background

A problem of Bollobás and Erdős, described in [Va99] as 'motivated by the task of generating a random triangle-free graph'. In [Bo98] it says they asked this at the 'Quo Vadis, Graph Theory?' conference in Fairbanks, Alaska, in 1990, 'while admiring the playful bears'.

Grable [Gr97] proved that, for every ϵ>0\epsilon>0,

P(f(n)>n7/4+ϵ)→0.\mathbb{P}(f(n)>n^{7/4+\epsilon})\to 0.

Bohman, Frieze, and Lubetzky [BFL15] proved that f(n)=n3/2+o(1)f(n)=n^{3/2+o(1)} almost surely - in other words, for every ϵ>0\epsilon>0,

P(n3/2−ϵ<f(n)<n3/2+ϵ)→1.\mathbb{P}(n^{3/2-\epsilon}<f(n)<n^{3/2+\epsilon})\to 1.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1155 link

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