Minimum number of edges that occur in odd cycles

Abstract

If a graph G has n ≥ 4k vertices and more than n 2 /4 edges, then it contains a copy of C 2k+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 n/2 + 1, and this bound is tight. They also showed that the minimum number of edges in G that occur in a copy of C 2k+1 for k ≥ 2 suddenly starts being quadratic in n, and conjectured that for any k ≥ 2, the correct lower bound should be 2n 2 /9 − O(n). Very recently, Füredi and Maleki constructed a counterexample for k = 2 and proved an asymptotically matching lower bound, namely that for any ε > 0 graphs with (1 + ε)n 2 /4 edges contain at least (2 + √ 2)n 2 /16 ∼ 0.2134n 2 edges that occur in C 5 . 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 2 /4 +1 edges has at least (2+ √ 2)n 2 /16− O(n 15/8 ) edges that occur in C 5 . Next, for all k ≥ 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 n 2 4 + 1 − n+4 6 n+1 6 = 2n 2 /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 answers 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 the existence of a single copy of C 2k+1 , which a standard flag algebra approach would not be able to handle. Also, we establish the first application of the semidefinite method in a setting, where the set of tight examples has exponential size and arises from two very different constructions.

The Minimum Number of Triangular Edges and a Symmetrization Method for Multiple Graphs

“…The second part of our Conjecture 1, namely that the extremal graph should be from a G(a, b, c), is still open. Grzesik, P. Hu, and Volec [7] using Razborov's flag algebra method showed that every nvertex graph with n 2 /4 + 1 edges has at least (n 2 /4) − n 2 /8(2 + √ 2) − εn 2 pentagonal edges for n > n 0 (ε) for every ε > 0. They also proved that those graphs have at most n 2 /36 + εn 2 C 2k+1 -edges for n > n k (ε) for every ε > 0 and k ≥ 3.…”

Minimizing the Number of Triangular Edges

“…We believe that the method of Grzesik, Hu and Volec [6] should be sufficient to give the exact smallest number of C 2k+1 -edges in a graph with n vertices and e edges, for any fixed k 2, provided that n is sufficiently large. Furthermore, their stability result should be sufficient to establish that, for sufficiently large n, the construction described earlier is the unique extremal construction.…”

“…Very recently, more progress on Problem 7.2 was made by Grzesik, Hu and Volec [6]. For any fixed k 2, they obtained asymptotically sharp bounds for the smallest possible number of C 2k+1 -edges in a graph with n vertices and at least n 2 /4 + 1 edges, using the method of flag algebras.…”

Edges not in any monochromatic copy of a fixed graph

“…A less studied but still quite natural question is to maximise the number of edges that do not belong to any forbidden subgraph. Such problems in the Turán context (where we are given the order n and the size m > ex(n, H) of a graph G) were studied in [13,19,22,23]. In the Ramsey context, a problem of this type seems to have been first posed by Erdős, Rousseau, and Schelp (see [12,Page 84]).…”