Minimum number of edges that occur in odd cycles

Abstract

If a graph 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 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 a counterexample for $k=2$ and proved asymptotically matching lower bound, namely that for any $\varepsilon>0$ graphs with $(1+\varepsilon)n^2/4$ edges contain at least $(2+\sqrt{2})n^2/16 \approx 0.2134n^2$ edges that occur in $C_5$. In this paper, we use a different approach to tackle this problem and obtain the following stronger result: Any $n$-vertex graph with at least $\lfloor n^2/4\rfloor+1$ edges has at least $(2+\sqrt{2})n^2/16-O(n^{15/8})$ edges that occur in $C_5$. Next, for all $k\ge 3$ and $n$ sufficiently large, we determine the exact minimum number of edges that occur in $C_{2k+1}$ for $n$-vertex graphs with more than $n^2/4$ edges, and show it is indeed equal to $\lfloor\frac{n^2}4\rfloor+1-\lfloor\frac{n+4}6\rfloor\lfloor\frac{n+1}6\rfloor=2n^2/9-O(n)$. For both results, we give a structural description of the extremal configurations as well as obtain the corresponding stability results, which answer a conjecture of F\"uredi and Maleki. The main ingredient is a novel approach that combines the flag algebras together with ideas from finite forcibility of graph limits. This approach allowed us to keep track of the extra edge needed to guarantee an existence of a $C_{2k+1}$. Also, we establish the first application of semidefinite method in a setting, where the set of tight examples has exponential size, and arises from different constructions.