Extremal Graphs for Blow-Ups of Keyrings

Ni, Zhenyu; Kang, Lei; Zhu, Hui June

doi:10.1007/s00373-020-02203-7

Cited by 11 publications

(5 citation statements)

References 9 publications

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“…Since δ(K q−1 ∨F ) ≤ ρ(K q−1 ∨F ) ≤ ∆(K q−1 ∨F ), in view of the construction of K q−1 ∨ F , we see that ρ(K q−1 ∨ F ) = Θ(n). Combining (11) and the fact y T y ≤ n, there exists a constant c 1 > 0 such that ρ(K q−1 ∨ F ′ ) − ρ(K q−1 ∨ F ) ≥ c 1 n , and this implies that ρ(K q−1 ∨ T p (n − q + 1) ≥ ρ(K q−1 ∨ F ′ ). Therefore,…”

Section: Proof Of Theorem 11mentioning

confidence: 88%

School Mathematics Textbooks in China

Wang¹

2015

East China Normal University Scientific Reports

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Let µ and ν be two probability measures on R d , where µ(dx) = e −V (x) dx R d e −V (x) dx for some V ∈ C 1 (R d ). Explicit sufficient conditions on V and ν are presented such that µ * ν satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, if V (x) = λ|x| 2 for some λ > 0 and ν(e λθ|•| 2 ) < ∞ for some θ > 1, then µ * ν satisfies the log-Sobolev inequality. This improves and extends the recent results on the log-Sobolev inequality derived in [20] for convolutions of the Gaussian measure and compactly supported probability measures. On the other hand, it is well known that the log-Sobolev inequality for µ * ν implies ν(e ε|•| 2 ) < ∞ for some ε > 0. RésuméDes conditions explicites suffisantes sur V et ν sont présentées telles que µ * ν satisfait des inégalités de Sobolev logarithmique, de Poincaré et de super-Poincaré. En particulier, si V (x) = λ|x| 2 pour quelque λ > 0 et ν(e λθ|•| 2 ) < ∞ avec θ > 1, alors µ * ν satisfait l'inégalité de Sobolev logarithmique. Cela améliore et étend des résultats récents sur l'inégalité de Sobolev logarithmique obtenus dans [20] pour des convolutions de la mesure de Gauss et des mesures de probabilité à support compact. D'autre part, il est bien connu que l'inégalité de Sobolev logarithmique pour µ * ν implique ν(e ε|•| 2 ) < ∞ pour quelque ε > 0.

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Section: Proof Of Theorem 11mentioning

confidence: 88%

School Mathematics Textbooks in China

Wang¹

2015

East China Normal University Scientific Reports

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“…. So far, Turán number of the edge blow-up of many families of graphs are known, for example, trees, cycles, keyrings, cliques K r with ≥ p r + 1, and complete bipartite graphs K s t , with ≥ p 3, see [3,12,13,17,22]. In general, a remarkable result by Yuan [22] gives the range of n G ex( , ) p+1 for ≥ p χ G ( ) + 1.…”

Section: +1mentioning

confidence: 99%

“…One can see

{F}_{k}={K}_{1,k}^{3}

and

{K}_{1}\otimes k{K}_{p}={K}_{1,k}^{p+1}

. So far, Turán number of the edge blow‐up of many families of graphs are known, for example, trees, cycles, keyrings, cliques

{K}_{r}

with

p\ge r+1

, and complete bipartite graphs

{K}_{s,t}

with

p\ge 3

, see [3, 12, 13, 17, 22]. In general, a remarkable result by Yuan [22] gives the range of

\text{ex}(n,{G}^{p+1})

for

p\ge \chi (G)+1

.…”

Section: Introductionmentioning

confidence: 99%

Turán number of the odd‐ballooning of complete bipartite graphs

Peng,

Xia

2024

Journal of Graph Theory

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Given a graph , the Turán number is the maximum possible number of edges in an ‐vertex ‐free graph. The study of Turán number of graphs is a central topic in extremal graph theory. Although the celebrated Erdős‐Stone‐Simonovits theorem gives the asymptotic value of for nonbipartite , it is challenging in general to determine the exact value of for . The odd‐ballooning of is a graph such that each edge of is replaced by an odd cycle and all new vertices of odd cycles are distinct. Here the length of odd cycles is not necessarily equal. The exact value of Turán number of the odd‐ballooning of is previously known for being a cycle, a path, a tree with assumptions, and . In this paper, we manage to obtain the exact value of Turán number of the odd‐ballooning of with , where and each odd cycle has length at least five.

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“…We would like to point out that in [10] Ni, Kang and Shan have determined the extremal graphs for C p+1 k, 1 (see Theorem 1.2). In this paper we suppose ℓ ≥ 2.…”

Section: Simonovits Proposed the Following Problemmentioning

confidence: 99%

“…In some way it is a generalization of cycle and star. Ni, Kang and Shan [10] determined the extremal graphs for edge blow-up of keyrings. A lollipop C k, ℓ is the graph obtained from a cycle of order k by appending a path P ℓ+1 to one of its vertices, and the vertex v ∈ V (C k, ℓ ) of degree 3 is called the center of the lollipop.…”

Section: Introductionmentioning

confidence: 99%

Extremal graphs for edge blow-up of lollipops

Zhai¹,

Yuan²,

Ni³

2022

Preprint

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Given a graph H and an integer p (p ≥ 2), the edge blow-up H p+1 of H is the graph obtained from replacing each edge in H by a clique of order (p + 1), where the new vertices of the cliques are all distinct. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. Very recently some substantial progress of the extremal graphs for H p+1 of larger p has been made by Yuan. The range of Turán numbers for edge blow-up of all bipartite graphs when p ≥ 3 and the exact Turán numbers for edge blow-up of all non-bipartite graphs when p ≥ χ(H) + 1 has been determined by Yuan (2022), where χ(H) is the chromatic number of H. A lollipop C k, ℓ is the graph obtained from a cycle C k by appending a path P ℓ+1 to one of its vertices. In this paper, we consider the extremal graphs for C p+1 k, ℓ of the rest cases p = 2 and p = 3.

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Extremal Graphs for Blow-Ups of Keyrings

Cited by 11 publications

References 9 publications

School Mathematics Textbooks in China

School Mathematics Textbooks in China

Turán number of the odd‐ballooning of complete bipartite graphs

Extremal graphs for edge blow-up of lollipops

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