Minimum number of edges that occur in odd cycles

If a graph G has n ≥ 4k vertices and more than n/4 edges, then it contains a copy of C2k+1. In 1992, Erdős, Faudree and Rousseau showed even more, that the number of edges that occur in a triangle of such a G is at least 2 bn/2c+ 1, and this bound is tight. They also showed that the minimum number of edges in G that occur in a copy of C2k+1 for k ≥ 2 suddenly starts being quadratic in n, and conjectured that for any k ≥ 2, the correct lower bound should be 2n/9 − O(n). Very recently, Füredi and Maleki constructed a counterexample for k = 2 and proved asymptotically matching lower bound, namely that for any ε > 0 graphs with (1+ε)n/4 edges contain at least (2 + √ 2)n/16 ∼ 0.2134n edges that occur in C5. In this paper, we use a different approach to tackle this problem and prove the following stronger result: Every n-vertex graph with at least ⌊ n/4 ⌋ +1 edges has at least (2+ √ 2)n/16− O(n) edges that occur in C5. Next, for all k ≥ 3 and n sufficiently large, we determine the exact minimum number of edges that occur in C2k+1 for n-vertex graphs with more than n /4 edges, and show it is indeed equal to ⌊ n 4 ⌋ + 1 − ⌊ n+4 6 ⌋ ⌊ n+1 6 ⌋ = 2n/9 − O(n). For both of these results, we give a structural description of all the large extremal configurations as well as obtain the corresponding stability results, which answer a conjecture of Füredi and Maleki. The main ingredient of our results is a novel approach that combines the semidefinite method from flag algebras together with ideas from finite forcibility of graph limits, which may be of independent interest. This approach allowed us to keep track of the additional extra edge needed to guarantee even an existence of a single copy of C2k+1, which a standard flag algebra approach would not be able to handle. Also, we establish the first application of semidefinite method in a setting, where the set of tight examples has exponential size, and arises from two very different constructions.