If a graph $G$ has $n/ge4k$ vertices and more than $n^2/4$ edges, then it contains a copy of $C_{2k+1}$. In 1992, Erd/H{o}s, Faudree and Rousseau showed even more, that the number of edges that occur in a triangle of such a $G$ is at least $2/lfloor n/2/rfloor+1$, and this bound is tight. They also showed that the minimum number of edges that occur in a $C_{2k+1}$ for $k/ge2$ is at least $11n^2/144-O(n)$, and conjectured that for any $k/ge2$, the correct lower bound should be $2n^2/9-O(n)$.

Very recently, F/"uredi and Maleki constructed $n$-vertex graphs with more than $n^2/4$ edges such that only $(2+/sqrt{2}+o(1))n^2/16/approx0.213n^2$ of them occur in $C_5$, which disproves the conjecture for $k=2$. They also proved that this construction is asymptotically best possible by showing that for any $/varepsilon>0$, graphs with $(1/4+/varepsilon)n^2$ edges contain at least $(2+/sqrt{2}-/varepsilon)n^2/16$ edges that occur in $C_5$.

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