A note on the maximum number of triangles in a <i>C</i><sub>5</sub>‐free graph

Ergemlidze, Beka; Methuku, Abhishek; Salia, Nika; Győri, Ervin

doi:10.1002/jgt.22390

Cited by 36 publications

(22 citation statements)

References 5 publications

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“…The systematic study of the function ex(n, H, F ) was initiated by Alon and Shikhelman in [2], where they improved the result of Bollobás and Győri by showing that ex(n, C 3 , C 5 ) ≤ (1 + o(1)) √ 3 2 n 3/2 . This bound was further improved in [12] and then very recently in [13] by Ergemlidze and Methuku who showed that ex(n, C 3 , C 5 ) < (1 + o(1))0.232n 3/2 . Another notable result is the exact computation of ex(n, C 5 , C 3 ) by Hatami, Hladký, Král, Norine, and Razborov [28] and independently by Grzesik [21], where they showed that it is equal to ( n 5 ) 5 .…”

Section: Generalized Turán Problemsmentioning

confidence: 90%

Generalized Turán problems for even cycles

Gerbner

Győri

Methuku

et al. 2020

Journal of Combinatorial Theory, Series B

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Given a graph H and a set of graphs F , let ex(n, H, F) denote the maximum possible number of copies of H in an F-free graph on n vertices. We investigate the function ex(n, H, F), when H and members of F are cycles. Let C k denote the cycle of length k and let C k = {C 3 , C 4 ,. .. , C k }. We highlight the main results below. (i) We show that ex(n, C 2l , C 2k) = Θ(n l) for any l, k ≥ 2. Moreover, in some cases we determine it asymptotically: We show that ex(n, C 4 , C 2k) = (1 + o(1)) (k−1)(k−2) 4 n 2 and that the maximum possible number of C 6 's in a C 8-free bipartite graph is n 3 + O(n 5/2). (ii) Erdős's Girth Conjecture states that for any positive integer k, there exist a constant c > 0 depending only on k, and a family of graphs {G n } such that |V (G n)|= n, |E(G n)|≥ cn 1+1/k with girth more than 2k. Solymosi and Wong [38] proved that if this conjecture holds, then for any l ≥ 3 we have ex(n, C 2l , C 2l−1) = Θ(n 2l/(l−1)). We prove that their result is sharp in the sense that forbidding any other even cycle decreases the number of C 2l 's significantly: For any k > l, we have ex(n, C 2l , C 2l−1 ∪ {C 2k }) = Θ(n 2). More generally, we show that for any k > l and m ≥ 2 such that 2k = ml, we have ex(n, C ml , C 2l−1 ∪ {C 2k }) = Θ(n m). (iii) We prove ex(n, C 2l+1 , C 2l) = Θ(n 2+1/l), provided a stronger version of Erdős's Girth Conjecture holds (which is known to be true when l = 2, 3, 5). This result is also sharp in the sense that forbidding one more cycle decreases the number of C 2l+1 's significantly: More precisely, we have ex(n, C 2l+1 , C 2l ∪ {C 2k }) = O(n 2− 1 l+1), and ex(n, C 2l+1 , C 2l ∪ {C 2k+1 }) = O(n 2) for l > k ≥ 2. (iv) We also study the maximum number of paths of given length in a C k-free graph, and prove asymptotically sharp bounds in some cases.

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Section: Generalized Turán Problemsmentioning

confidence: 90%

Generalized Turán problems for even cycles

Gerbner

Győri

Methuku

et al. 2020

Journal of Combinatorial Theory, Series B

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“…Erdős [8] first determined ex(n, K t , K r ) for all t < r. Bollobás and Győri [3] determined the order of magnitude of ex(n, C 3 , C 5 ). Their estimate was improved by Alon and Shikhelman [1] and recently by Ergemlidze, Győri, Methuku and Salia [10]. Alon and Shikhelman obtained a number of results on ex(n, T, H) for different T and H and posed several open problems in [1].…”

Section: Short Proofs Of Two Theorems Of Luomentioning

confidence: 97%

Extensions of the Erdős–Gallai theorem and Luo’s theorem

Ning¹,

Peng²

2019

Combinator. Probab. Comp.

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The famous Erdős-Gallai Theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least 2m n edges. In this note, we first establish a simple but novel extension of the Erdős-Gallai Theorem by proving that every graph G contains a path with at leastWe also construct a family of graphs which shows our extension improves the estimate given by Erdős-Gallai Theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with l vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.

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“…The constant in the upper bound was improved to √ 3 3 by Alon and Shikhelman [1] and by Ergemlidze, Győri, Methuku and Salia [10]. Győri and Li [23] give bounds on ex(n, K 3 , C 2k+1 ).…”

Section: Introductionmentioning

confidence: 99%

Counting copies of a fixed subgraph in F-free graphs

Gerbner

Palmer

2019

European Journal of Combinatorics

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Fix graphs F and H and let ex(n, H, F ) denote the maximum possible number of copies of the graph H in an n-vertex F -free graph. The systematic study of this function was initiated by Alon and Shikhelman [J. Comb. Theory, B. 121 (2016)]. In this paper, we give new general bounds concerning this generalized Turán function. We also determine ex(n, P k , K 2,t ) (where P k is a path on k vertices) and ex(n, C k , K 2,t ) asymptotically for every k and t. For example, it is shown that for t ≥ 2 and k ≥ 5 we have ex(n, C k , K 2,t ) = 1 2k + o(1) (t − 1) k/2 n k/2 . We also characterize the graphs F that cause the function ex(n, C k , F ) to be linear in n. In the final section we discuss a connection between the function ex(n, H, F ) and so-called Berge hypergraphs.

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A note on the maximum number of triangles in a C₅‐free graph

Abstract: We prove that the maximum number of triangles in a C 5‐free graph on n vertices is at most 122(1+o(1))n3∕2, improving an estimate of Alon and Shikhelman.

Cited by 36 publications

References 5 publications

Generalized Turán problems for even cycles

Generalized Turán problems for even cycles

Extensions of the Erdős–Gallai theorem and Luo’s theorem

Counting copies of a fixed subgraph in F-free graphs

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A note on the maximum number of triangles in a C5‐free graph

Abstract: We prove that the maximum number of triangles in a C 5‐free graph on n vertices is at most 122(1+o(1))n3∕2, improving an estimate of Alon and Shikhelman.

Cited by 36 publications

References 5 publications

Generalized Turán problems for even cycles

Generalized Turán problems for even cycles

Extensions of the Erdős–Gallai theorem and Luo’s theorem

Counting copies of a fixed subgraph in F-free graphs

Contact Info

Product

Resources

About

A note on the maximum number of triangles in a C₅‐free graph