# For Which Densities are Random Triangle-Free Graphs Almost Surely Bipartite?

Denote by $${\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)}$$ the class of all triangle-free graphs on n vertices and m edges. Our main result is the following sharp threshold, which answers the question for which densities a typical triangle-free graph is bipartite. Fix ε > 0 and let $$t_{3} = t_{3} {\left( n \right)}\frac{{{\sqrt 3 }}} {4}n^{{3/2}} {\sqrt {\log {\kern 1pt} {\kern 1pt} n} }$$. If n/2 ≤ m ≤ (1 − ε) t 3, then almost all graphs in $${\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)}$$ are not bipartite, whereas if m ≥ (1 + ε)t 3, then almost all of them are bipartite. For m ≥ (1 + ε)t 3, this allows us to determine asymptotically the number of graphs in $${\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)}$$. We also obtain corresponding results for C -free graphs, for any cycle C of fixed odd length.

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Correspondence to Deryk Osthus*.

Forschergruppe “Algorithmen, Struktur, Zufall” supported by Deutsche Forschungsgemeinschaft grant FOR 413/1-1

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Osthus*, D., Prömel, H.J. & Taraz, A. For Which Densities are Random Triangle-Free Graphs Almost Surely Bipartite?. Combinatorica 23, 105–150 (2003). https://doi.org/10.1007/s00493-003-0016-1

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• 05C80
• 05C35
• 05A16