An Erdős–Gallai type theorem for uniform hypergraphs

Davoodi, Akbar; Győri, Ervin; Methuku, Abhishek; Tompkins, Casey

doi:10.1016/j.ejc.2017.10.006

Cited by 40 publications

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“…The blow-up of a C 5 gives a lower-bound of ( n 5 ) 5 when n is divisible by 5. Hatami, Hladký, Král', Norine and Razborov [24] and independently Grzesik [19] proved ex(n, C 5 , K 3 ) ≤ n 5 5 .…”

Section: Introductionmentioning

confidence: 96%

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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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Section: Introductionmentioning

confidence: 96%

“…For example, (2) follows by applying Corollary 1 in the case when H = K t . Applying (1) to the case when H = C 5 gives ex(n, C 5 , F ) ≤ n 5 5 + o(n 5 )…”

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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“…Results of Győri, Katona, and Lemons [19] and Davoodi, Győri, Methuku and Tompkins [4] establish an analogue of the Erdős-Gallai theorem for Berge paths. Győri and Lemons [20] proved that the maximum number of hypereges in an n-vertex r-uniform Berge-C 2k -free hypergraph (for r ≥ 3) is O(n 1+1/k ).…”

Section: Introductionmentioning

confidence: 84%

“…For the case k = r + 1, Győri, Katona and Lemons conjectured that the upper bound should have the same form as the k > r + 1 case. This was settled by Davoodi, Győri, Methuku and Tompkins [4] who showed that if k = r + 1 > 2, then ex r (n, Berge-P k ) ≤ n k k r = n.…”

Section: Berge Treesmentioning

confidence: 99%

General lemmas for Berge–Turán hypergraph problems

Gerbner

Methuku

Palmer

2020

European Journal of Combinatorics

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“…The bounds in Theorems 1.1 and 1.2 are best possible for infinitely many n and k. The theorems were refined in [4,6,10,11,12]. Recently, Győri, Katona, and Lemons [8] and Davoodi, Győri, Methuku, and Tompkins [1] extended Theorem 1.1 to r-uniform hypergraphs (r-graphs, for short). They considered Berge paths and cycles.…”

Section: Introductionmentioning

confidence: 99%

On r-uniform hypergraphs with circumference less than r

Kostochka

Luo

2020

Discrete Applied Mathematics

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We show that for each k ≥ 4 and n > r ≥ k + 1, every n-vertex r-uniform hypergraph with no Berge cycle of length at least k has at most (k−1)(n−1) r edges. The bound is exact, and we describe the extremal hypergraphs. This implies and slightly refines the theorem of Győri, Katona and Lemons that for n > r ≥ k ≥ 3, every n-vertex r-uniform hypergraph with no Berge path of length k has at most (k−1)n r+1 edges. To obtain the bounds, we study bipartite graphs with no cycles of length at least 2k, and then translate the results into the language of multi-hypergraphs.Mathematics Subject Classification: 05C35, 05C38.Later, the remaining case k = r + 1 was resolved by Davoodi, Győri, Methuku, and Tompkins.Theorem 1.5 (Davoodi, Győri, Methuku and Tompkins [1]). Let k = r + 1 > 2, and let H be an n-vertex r-graph with no Berge-path of length k. Then e(H) ≤ n.Furthermore, the bounds in these three theorems are sharp for each k and r for infinitely many n.Very recently, several interesting results were obtained for Berge paths and cycles for k ≥ r + 1. First, Győri, Methuku, Salia, Tompkins, and Vizer [9] proved an asymptotic version of the Erdős-Gallai theorem for Berge-paths in connected hypergraphs whenever r is fixed and n and k tend to infinity. For Berge-cycles, the exact result for k ≥ r + 3 was obtained in [5]: Theorem 1.6 (Füredi, Kostochka and Luo [5]). Let k ≥ r + 3 ≥ 6, and let H be an n-vertex r-graph with no Berge-cycles of length k or longer. Then e(H) ≤ n−1 k−2 k−1 r . This theorem is a hypergraph version of Theorem 1.2 and an analog of Theorem 1.4. It also somewhat refines Theorem 1.4 for k ≥ r + 3. Later, Ergemlidze, Győri, Methuku, Salia, Thompkins, and Zamora [3] extended the results to to k ∈ {r + 1, r + 2}: Theorem 1.7 (Ergemlidze et al. [3]). If k ≥ 4 and H is an n-vertex r-graph with no Berge-cycles of length k or longer, then k = r + 1 and e(H) ≤ n − 1, or k = r + 2 and e(H) ≤ n−1 k−2 k−1 r .The goal of this paper is to prove a hypergraph version of Theorem 1.2 for r-graphs with no Berge-cycles of length k or longer in the case k ≤ r. Our result is an analog of Theorem 1.3 and yields a refinement of it.Our approach is to consider bipartite graphs in which the vertices in one of the parts have degrees at least r, and to analyze the structure of such graphs with circumference less than 2k. After that, we apply the obtained results to the incidence graphs of r-uniform hypergraphs. In this way, our methods differ from those of [8],[1], [9], and [5]. Notation and results Hypergraph notationThe lower rank of a multi-hypergraph H is the size of a smallest edge of H.In view of the structure of our proof, it is more convenient to consider hypergraphs with lower rank at least r instead of r-uniform hypergraphs. It also yields formally stronger statements of the results.The incidence graph G(H) of a multi-hypergraph H = (V, E) is the bipartite graph with parts V and E where v ∈ V is adjacent to e ∈ E iff in H vertex v belongs to edge e.There are several versions of connectivity of hypergraphs. We will call a m...

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An Erdős–Gallai type theorem for uniform hypergraphs

Cited by 40 publications

References 3 publications

Counting copies of a fixed subgraph in F-free graphs

Counting copies of a fixed subgraph in F-free graphs

General lemmas for Berge–Turán hypergraph problems

On r-uniform hypergraphs with circumference less than r

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