Extremal Graphs for Intersecting Triangles

Erdös, Paul; Füredi, Zoltán; Gould, Ronald J.; Gunderson, David S.

doi:10.1006/jctb.1995.1026

Cited by 96 publications

(86 citation statements)

References 11 publications

(13 reference statements)

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“…The following lemma is proved in . Lemma Let H be a graph and b a nonnegative integer such that

b \leq Δ (H) - 2

, and let

ν = ν (H), Δ = Δ (H)

.…”

Section: Several Technical Lemmasmentioning

confidence: 99%

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Extremal graphs for the k‐flower

Yuan

2018

Journal of Graph Theory

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The Turán number of a graph H, ex(n,H), is the maximum number of edges in any graph of order n that does not contain an H as a subgraph. A graph on 2k+1 vertices consisting of k triangles that intersect in exactly one common vertex is called a k‐fan, and a graph consisting of k cycles that intersect in exactly one common vertex is called a k‐flower. In this article, we determine the Turán number of any k‐flower containing at least one odd cycle and characterize all extremal graphs provided n is sufficiently large. Erdős, Füredi, Gould, and Gunderson determined the Turán number for the k‐fan. Our result is a generalization of their result. The addition aim of this article is to draw attention to a powerful tool, the so‐called progressive induction lemma of Simonovits.

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“…The following lemma is proved in . Lemma Let H be a graph and b a nonnegative integer such that

b \leq Δ (H) - 2

, and let

ν = ν (H), Δ = Δ (H)

.…”

Section: Several Technical Lemmasmentioning

confidence: 99%

“…Proof We only prove the lemma for

k_{1} > k_{2}

, since for

k_{1} = k_{2}

, the lemma is the Lemma 6.2 in without determining the extremal graphs. Observe that

G_{c r}

is a bipartite graph, and

false| V_{0} false| \cdot false| V_{1} false| - e false(G_{c r} false)

is the number of edges missing from the complete bipartite graph.…”

Section: Several Technical Lemmasmentioning

confidence: 99%

Extremal graphs for the k‐flower

Yuan

2018

Journal of Graph Theory

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“…. , d n ) be a non-increasing graphic degree sequence with with degree sum at least k(2n − k − 1) + 2 and n > k 2 …”

Section: The Main Theoremmentioning

confidence: 99%

Graphic Sequences with a Realization Containing a Union of Cliques

Ferrara

2007

Graphs and Combinatorics

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For any simple graph H, let σ(H, n) be the minimum m so that for any realizable degree sequence π = (d1, d2, . . . , dn) with sum of degrees at least m, there exists an n-vertex graph G witnessing π that contains H as a weak subgraph. Let F k denote the friendship graph on 2k + 1 vertices, that is, the graph of k triangles intersecting in a single vertex. In this paper, for n sufficiently large, σ(F k , n) is determine precisely.

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“…In 1995, Erdős et al. have determined the value of the function ex

(n, F_{k})

as well as the

F_{k}

‐extremal graphs for every fixed k and whenever n is large. They have proved the following result.…”

Section: Introductionmentioning

confidence: 99%

Decompositions of Graphs into Fans and Single Edges

Liu

Sousa²

2016

Journal of Graph Theory

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Given two graphs G and H, an H‐decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. Let φ(n,H) be the smallest number ϕ such that any graph G of order n admits an H‐decomposition with at most ϕ parts. Pikhurko and Sousa conjectured that φ(n,H)= ex (n,H) for χ(H)≥3 and all sufficiently large n, where ex (n,H) denotes the maximum number of edges in a graph on n vertices not containing H as a subgraph. Their conjecture has been verified by Özkahya and Person for all edge‐critical graphs H. In this article, the conjecture is verified for the k‐fan graph. The k‐fan graph, denoted by Fk, is the graph on 2k+1 vertices consisting of k triangles that intersect in exactly one common vertex called the center of the k‐fan.

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Extremal Graphs for Intersecting Triangles

Abstract: It is known that for a graph on n vertices n 2 /4 + 1 edges is sufficient for the existence of many triangles. In this paper, we determine the minimum number of edges sufficient for the existence of k triangles intersecting in exactly one common vertex.

Cited by 96 publications

References 11 publications

Extremal graphs for the k‐flower

Extremal graphs for the k‐flower

Graphic Sequences with a Realization Containing a Union of Cliques

Decompositions of Graphs into Fans and Single Edges

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