The Inducibility of Graphs on Four Vertices

Hirst, James

doi:10.1002/jgt.21733

Cited by 40 publications

(43 citation statements)

References 20 publications

(56 reference statements)

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“…For each i ∈ [m], we set b i := |V i |/n. By (24), the continuity of λ(B(·)), and the compactness of S m , we can assume that the vector b = (b 1 , ... , b m ) is c 3 -close to a maximiser a of λ(B(·)), that is,…”

Section: Main Results For Perfect Stabilitymentioning

confidence: 99%

Strong forms of stability from flag algebra calculations

Pikhurko

Sliacan

Tyros

2019

Journal of Combinatorial Theory, Series B

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Given a hereditary family G of admissible graphs and a function λ(G) that linearly depends on the statistics of order-κ subgraphs in a graph G, we consider the extremal problem of determining λ(n, G), the maximum of λ(G) over all admissible graphs G of order n. We call the problem perfectly B-stable for a graph B if there is a constant C such that every admissible graph G of order n C can be made into a blow-up of B by changing at most C(λ(n, G)−λ(G)) n 2 adjacencies. As special cases, this property describes all almost extremal graphs of order n within o(n 2 ) edges and shows that every extremal graph of order n n 0 is a blow-up of B.We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results.

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Section: Main Results For Perfect Stabilitymentioning

confidence: 99%

Strong forms of stability from flag algebra calculations

Pikhurko

Sliacan

Tyros

2019

Journal of Combinatorial Theory, Series B

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“…Problems of maximizing the number of induced copies of a fixed small graph H have attracted a lot of attention recently [8,14,29]. For a list of other results on this so called inducibility of small graphs of order up to 5, see the work of Even-Zohar and Linial [8].…”

Section: Introductionmentioning

confidence: 99%

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

Balogh

Lidický

et al. 2016

European Journal of Combinatorics

110

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“…Flag algebras have been very successful in tackling various problems. To mention some of them: Caccetta-Häggkvist conjecture [21,25,36], various Turán-type problems in graphs [10,20,22,24,29,31,32,34,37,39], hypergraphs [3,15,16,19,30] and hypercubes [2,5], extremal problems in a colored environment [4,9,23,26] and also to problems in geometry [27] or extremal theory of permutations [6]. For more details on these applications, see a survey of Razborov [35].…”

Section: Flag Algebrasmentioning

confidence: 99%

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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The Inducibility of Graphs on Four Vertices

Cited by 40 publications

References 20 publications

Strong forms of stability from flag algebra calculations

Strong forms of stability from flag algebra calculations

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

Minimum number of edges that occur in odd cycles

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