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

Füredi, Zoltán; Maleki, Zeinab

doi:10.1017/s0963548316000407

Cited by 7 publications

(15 citation statements)

References 9 publications

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“…In the case of odd cycles of length at least 7 and n sufficiently large, we determine the exact value of the number of edges that occur in C 2k+1 , which indeed matches the value given by Construction 1, which answers another conjecture of Füredi and Maleki [17,Conjecture 8].…”

Section: Introductionsupporting

confidence: 55%

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Minimum number of edges that occur in odd cycles

Grzesik

Volec

2019

Journal of Combinatorial Theory, Series B

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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.

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Section: Introductionsupporting

confidence: 55%

“…Füredi and Maleki [18] determined an asymptotically optimal lower bound for this problem. Note that the corresponding approximate result for triangles was proven by Füredi and Maleki in [17].…”

Section: Discussionmentioning

confidence: 69%

Minimum number of edges that occur in odd cycles

Grzesik

Volec

2019

Journal of Combinatorial Theory, Series B

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“…Füredi and Maleki [4] proved an approximate version of the latter statement, which reads as follows.…”

Section: Introductionmentioning

confidence: 92%

“…Füredi and Maleki [4] conjectured that the minimizers of the number of triangular edges are graphs of the form G(a, b, c), or subgraphs of such graphs. Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 99%

Minimizing the Number of Triangular Edges

Gruslys

Letzter

2017

Combinator. Probab. Comp.

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“…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]).…”

Section: Introductionmentioning

confidence: 99%