On the cover Turán number of Berge hypergraphs

Abstract The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

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The formula for Turán number of spanning linear forests

Discrete Mathematics ◽

10.1016/j.disc.2020.111924 ◽

2020 ◽

Vol 343 (8) ◽

pp. 111924

Author(s):

Bo Ning ◽

Jian Wang

Keyword(s):

Turán Number

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An Ordered Turán Problem for Bipartite Graphs

The Electronic Journal of Combinatorics ◽

10.37236/2471 ◽

2012 ◽

Vol 19 (4) ◽

Author(s):

Craig Timmons

Keyword(s):

Complete Bipartite Graph ◽

Bipartite Graphs ◽

Free Graph ◽

The Family ◽

Turan Problem ◽

Vertex Graph ◽

Turán Number ◽

Turán Problem ◽

Natural Way

Let $F$ be a graph. A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. The Turán number of $F$, written $\textrm{ex}(n,F)$, is the maximum number of edges in an $F$-free graph with $n$ vertices. The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem. In this paper we introduce an ordered version of the Turán problem for bipartite graphs. Let $G$ be a graph with $V(G) = \{1, 2, \dots , n \}$ and view the vertices of $G$ as being ordered in the natural way. A zig-zag $K_{s,t}$, denoted $Z_{s,t}$, is a complete bipartite graph $K_{s,t}$ whose parts $A = \{n_1 < n_2 < \dots < n_s \}$ and $B = \{m_1 < m_2 < \dots < m_t \}$ satisfy the condition $n_s < m_1$. A zig-zag $C_{2k}$ is an even cycle $C_{2k}$ whose vertices in one part precede all of those in the other part. Write $\mathcal{Z}_{2k}$ for the family of zig-zag $2k$-cycles. We investigate the Turán numbers $\textrm{ex}(n,Z_{s,t})$ and $\textrm{ex}(n,\mathcal{Z}_{2k})$. In particular we show $\textrm{ex}(n, Z_{2,2}) \leq \frac{2}{3}n^{3/2} + O(n^{5/4})$. For infinitely many $n$ we construct a $Z_{2,2}$-free $n$-vertex graph with more than $(n - \sqrt{n} - 1) + \textrm{ex} (n,K_{2,2})$ edges.

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A Turán Type Problem Concerning the Powers of the Degrees of a Graph

The Electronic Journal of Combinatorics ◽

10.37236/1525 ◽

2000 ◽

Vol 7 (1) ◽

Cited By ~ 5

Author(s):

Yair Caro ◽

Raphael Yuster

Keyword(s):

Positive Integer ◽

Lower Bounds ◽

Degree Sequence ◽

Upper And Lower Bounds ◽

Exact Results ◽

Type Problem ◽

Maximum Value ◽

Turán Number

For a graph $G$ whose degree sequence is $d_{1},\ldots ,d_{n}$, and for a positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph $H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $t_{1}(n,H)$ is twice the Turán number of $H$. In this paper we consider the case $p>1$. For some graphs $H$ we obtain exact results, for some others we can obtain asymptotically tight upper and lower bounds, and many interesting cases remain open.

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On the Turán Number of Forests

The Electronic Journal of Combinatorics ◽

10.37236/3142 ◽

2013 ◽

Vol 20 (2) ◽

Cited By ~ 8

Author(s):

Hong Liu ◽

Bernard Lidicky ◽

Cory Palmer

Keyword(s):

Arbitrary Order ◽

Extremal Graph ◽

Extremal Graphs ◽

Turán Number

The Turán number of a graph $H$, $\mathrm{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. We determine the Turán number and find the unique extremal graph for forests consisting of paths when $n$ is sufficiently large. This generalizes a result of Bushaw and Kettle [Combinatorics, Probability and Computing 20:837--853, 2011]. We also determine the Turán number and extremal graphs for forests consisting of stars of arbitrary order.

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The Exact Turán Number of $F(3,3)$ and All Extremal Configurations

SIAM Journal on Discrete Mathematics ◽

10.1137/110841837 ◽

2013 ◽

Vol 27 (2) ◽

pp. 910-917

Author(s):

John Goldwasser ◽

Ryan Hansen

Keyword(s):

Turán Number

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The Turán Number Of The Fano Plane

COMBINATORICA ◽

10.1007/s00493-005-0034-2 ◽

2005 ◽

Vol 25 (5) ◽

pp. 561-574 ◽

Cited By ~ 37

Author(s):

Peter Keevash ◽

Benny Sudakov*

Keyword(s):

Fano Plane ◽

Turán Number

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On the Chromatic Thresholds of Hypergraphs

Combinatorics Probability Computing ◽

10.1017/s0963548315000061 ◽

2015 ◽

Vol 25 (2) ◽

pp. 172-212

Author(s):

JÓZSEF BALOGH ◽

JANE BUTTERFIELD ◽

PING HU ◽

JOHN LENZ ◽

DHRUV MUBAYI

Keyword(s):

Chromatic Number ◽

Minimum Degree ◽

Directed Graphs ◽

Open Problems ◽

Fibre Bundles ◽

Uniform Hypergraphs ◽

The Family ◽

Turán Number ◽

Chromatic Thresholds ◽

Extremal Set

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

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Stability and Turán Numbers of a Class of Hypergraphs via Lagrangians

Combinatorics Probability Computing ◽

10.1017/s0963548316000444 ◽

2017 ◽

Vol 26 (3) ◽

pp. 367-405 ◽

Cited By ~ 9

Author(s):

AXEL BRANDT ◽

DAVID IRWIN ◽

TAO JIANG

Keyword(s):

Positive Integer ◽

Structural Stability ◽

Pairwise Disjoint ◽

Large Family ◽

Extremal Graphs ◽

Uniform Hypergraphs ◽

Turán Number

Given a family ofr-uniform hypergraphs${\cal F}$(orr-graphs for brevity), the Turán number ex(n,${\cal F})$of${\cal F}$is the maximum number of edges in anr-graph onnvertices that does not contain any member of${\cal F}$. A pair {u,v} iscoveredin a hypergraphGif some edge ofGcontains {u, v}. Given anr-graphFand a positive integerp⩾n(F), wheren(F) denotes the number of vertices inF, letHFpdenote ther-graph obtained as follows. Label the vertices ofFasv1,. . .,vn(F). Add new verticesvn(F)+1,. . .,vp. For each pair of verticesvi, vjnot covered inF, add a setBi,jofr− 2 new vertices and the edge {vi, vj} ∪Bi,j, where theBi,jare pairwise disjoint over all such pairs {i, j}. We callHFpthe expanded p-clique with an embedded F. For a relatively large family ofF, we show that for all sufficiently largen, ex(n,HFp) = |Tr(n, p− 1)|, whereTr(n, p− 1) is the balanced complete (p− 1)-partiter-graph onnvertices. We also establish structural stability of near-extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Turán number is exactly determined (for largen).

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