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 _ℓ -free graphs, for any cycle C _ℓ of fixed odd length.