The inducibility of blow-up graphs

Hatami, Hamed; Hirst, James; Norine, Serguei

doi:10.48550/arxiv.1108.5699

Cited by 3 publications

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“…We are now able to define the inducibility of a graph H as It is not difficult to see that this limit always exists. The very recent work [7] gives some strong asymptotic results for a large class of graphs, but the problem of determining i(H) appears to be non-trivial even in some cases when H is a very small graph. Let H denote the complement graph of H. Note that we have that i(H) = i(H), so that we need only to consider one graph in a complementary pair.…”

Section: Introductionmentioning

confidence: 99%

The Inducibility of Graphs on Four Vertices

Hirst

2013

Journal of Graph Theory

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Abstract:We consider the problem of determining the maximum induced density of a graph H in any graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H. The exact value of this quantity is known only for a handful of small graphs and a specific set of complete multipartite graphs. Answering questions of Brown-Sidorenko and Exoo, we determine the inducibility of K 1,1,2 and the paw graph. The proof is obtained using semidefinite programming techniques based on a modern language of extremal graph theory, which we describe in full detail in an accessible setting. C 2013 Wiley Periodicals, Inc. J. Graph Theory 75: [231][232][233][234][235][236][237][238][239][240][241][242][243] 2014

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Section: Introductionmentioning

confidence: 99%

The Inducibility of Graphs on Four Vertices

Hirst

2013

Journal of Graph Theory

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“…Remark. See [17] for conditions under which inducibility is attained by blow-ups. Lemma 4's proof naturally leads to the following construction, by Pippenger and Golumbic [26].…”

Section: Lemma 4 For Every Twin-free Graph H and For Every Graph G R(...mentioning

confidence: 99%

A Note on the Inducibility of $$4$$ 4 -Vertex Graphs

Even-Zohar

Linial

2014

Graphs and Combinatorics

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There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph H is its extremal density, as an induced subgraph of G, where |G| → ∞. Already for 4-vertex graphs many questions are still open. Thus, the inducibility of the 4-path was addressed in a construction of Exoo (Ars Combin 22:5-10, 1986), but remains unknown. Refuting a conjecture of Erdős, Thomason (Combinatorica 17(1):125-134, 1997) constructed graphs with a small density of both 4-cliques and 4-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.

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“…The inducibility of a graph H is defined as lim n→∞ max G d(H; G), where the maximum is over all n-vertex graphs G. This natural parameter has been investigated for several types of graphs H. E.g., complete bipartite and multipartite graphs [1,2], very small graphs [7,13] and blow-up graphs [11].…”

Section: Introductionmentioning

confidence: 99%

On the 3‐Local Profiles of Graphs

Huang

Linial

Naves³

et al. 2013

Journal of Graph Theory

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For a graph G, let pi(G),i=0,...,3 be the probability that three distinct random vertices span exactly i edges. We call (p0(G),...,p3(G)) the 3‐local profile of G. We investigate the set scriptS3⊂double-struckR4 of all vectors (p0,...,p3) that are arbitrarily close to the 3‐local profiles of arbitrarily large graphs. We give a full description of the projection of S3 to the (p0,p3) plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality p0+p3≥14. We also give a full description of the triangle‐free case, i.e. the intersection of S3 with the hyperplane p3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

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The inducibility of blow-up graphs

Cited by 3 publications

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The Inducibility of Graphs on Four Vertices

The Inducibility of Graphs on Four Vertices

A Note on the Inducibility of $$4$$ 4 -Vertex Graphs

On the 3‐Local Profiles of Graphs

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