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

Denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)} $$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t_{3} = t_{3} {\left( n \right)}\frac{{{\sqrt 3 }}} {4}n^{{3/2}} {\sqrt {\log {\kern 1pt} {\kern 1pt} n} } $$\end{document}. If n/2 ≤ m ≤ (1 − ε) t3, then almost all graphs in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)} $$\end{document} are not bipartite, whereas if m ≥ (1 + ε)t3, then almost all of them are bipartite. For m ≥ (1 + ε)t3, this allows us to determine asymptotically the number of graphs in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)} $$\end{document}. We also obtain corresponding results for Cℓ-free graphs, for any cycle Cℓ of fixed odd length.