Turán numbers for odd wheels

Dzido, Tomasz; Jastrzȩbski, Andrzej

doi:10.1016/j.disc.2017.10.003

Cited by 17 publications

(10 citation statements)

References 3 publications

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“…Proof of Theorem 1.4: Sine ex(n, W 5 ) is determined in [2], we may suppose that k ≥ 3. Let L n be an extremal graph for W 2k+1 .…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…In this paper, we will consider the Turán numbers of wheels on odd numbers of vertices. Wheels on odd numbers of vertices are not contained in Simonovits' generalized theorem [11], since the decomposition family (see Section 3.2) of W 2k+1 does not contain any F ⊂ P t for some large t. In [2], Dzido and Jastrzȩbski determined ex(n, W 5 ) and ex(n, W 7 ) for all value of n. They also give a lower bound for general case. We will show that the lower bound in [2] is the exact value of ex(n, W 2k+1 ) for infinite value of n. Let n, n 0 and n 1 be integers.…”

Section: Introductionmentioning

confidence: 99%

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Extremal graphs for wheels

Yuan

2020

Preprint

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A wheel graph is a graph formed by connecting a single vertex to all vertices of a cycle. The extremal graphs for wheels on even number of vertices is determined by Simonovits in 1960s. In this paper, we determine the Turán numbers of wheels on odd number vertices. Wheels on odd numbers of vertices are the first cases that the extremal graphs are characterized when the decomposition families of graphs do not contain a linear forest.

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“…Proof of Theorem 1.4: Sine ex(n, W 5 ) is determined in [2], we may suppose that k ≥ 3. Let L n be an extremal graph for W 2k+1 .…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extremal graphs for wheels

Yuan

2020

Preprint

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“…. , C t by adding edges u i,j u i+1,j and u i,j u i+1,j+1 for all i ∈ [t − 1] and j ∈ [5], where all arithmetic on the index j + 1 here is done modulo 5, and finally adding two new non-adjacent vertices u and v such that u is adjacent to all vertices of C 1 and v is adjacent to all vertices of C t . The graph L t when t = 3 is depicted in Figure 3.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

Extremal $H$-Free Planar Graphs

Lan

Shi

Song

2019

Electron. J. Combin.

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Given a graph H, a graph is H-free if it does not contain H as a subgraph. We continue to study the topic of "extremal" planar graphs, that is, how many edges can an H-free planar graph on n vertices have? We define ex P (n, H) to be the maximum number of edges in an H-free planar graph on n vertices. We first obtain several sufficient conditions on H which yield ex P (n, H) = 3n − 6 for all n ≥ |V (H)|. We discover that the chromatic number of H does not play a role, as in the celebrated Erdős-Stone Theorem. We then completely determine ex P (n, H) when H is a wheel or a star. Finally, we examine the case when H is a (t, r)-fan, that is, H is isomorphic to K 1 + tK r−1 , where t ≥ 2 and r ≥ 3 are integers. However, determining ex P (n, H), when H is a planar subcubic graph, remains wide open.

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“…There is a partial result for the Turán number of odd wheels. Dzido and Jastrzȩbski [2] determined ex

(n, W_{5})

and ex

(n, W_{7})

for all value of

n

. They also established a lower bound on ex

(n, W_{2 k + 1})

for

k \geq 4

.…”

Section: Introductionmentioning

confidence: 99%

Extremal graphs for odd wheels

Yuan

2021

Journal of Graph Theory

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For a graph H, the Turán number of H, denoted by ex(n,H), is the maximum number of edges of an n‐vertex H‐free graph. Let g(n,H) denote the maximum number of edges not contained in any monochromatic copy of H in a 2‐edge‐coloring of Kn. A wheel Wm is a graph formed by connecting a single vertex to all vertices of a cycle of length m−1. The Turán number of W2k was determined by Simonovits in 1960s. In this paper, we determine ex(n,W2k+1) when n is sufficiently large. We also show that, for sufficient large n, g(n,W2k+1)=ex(n,W2k+1) which confirms a conjecture posed by Keevash and Sudakov for odd wheels.

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Turán numbers for odd wheels

Cited by 17 publications

References 3 publications

Extremal graphs for wheels

Extremal graphs for wheels

Extremal $H$-Free Planar Graphs

Extremal graphs for odd wheels

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