On the Evolution of Triangle-Free Graphs
Combinatorics, Probability and Computing
On extremal subgraphs of random graphs
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Extremal subgraphs of random graphs
Random Structures & Algorithms
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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) t3, then almost all graphs in $${\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)}$$ 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 $${\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)}$$. We also obtain corresponding results for Cℓ-free graphs, for any cycle Cℓ of fixed odd length.