scholarly journals The density Turan problem for 3-uniform linear hypertrees. An efficient testing algorithm

2018 ◽  
Vol 72 (2) ◽  
pp. 9
Author(s):  
Halina Bielak ◽  
Kamil Powroźnik
Keyword(s):  
Efficient Algorithm ◽  
Blow Up ◽  
Turan Problem ◽  
Turán Problem ◽  
Testing Algorithm

Let \(\mathcal{T}=(V,\mathcal{E})\) be a  3-uniform linear hypertree. We consider a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\). We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\) of the hypertree \(\mathcal{T}\), with hyperedge densities satisfying some conditions, such that the hypertree \(\mathcal{T}\) does not appear in a blow-up hypergraph as a transversal. We present an efficient algorithm to decide whether a given set of hyperedge densities ensures the existence of a 3-uniform linear hypertree \(\mathcal{T}\) in a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\).

The Density Turán Problem

2012 ◽  
Vol 21 (4) ◽  
pp. 531-553
Author(s):  
PÉTER CSIKVÁRI ◽  
ZOLTÁN LÓRÁNT NAGY
Keyword(s):  
Efficient Algorithm ◽  
Blow Up ◽  
Edge Density ◽  
Maximal Degree ◽  
Extremal Graphs ◽  
Graph Classes ◽  
Critical Edge ◽  
Turán Problem ◽  
Image Position ◽  
Extremal Structure

LetHbe a graph onnvertices and let the blow-up graphG[H] be defined as follows. We replace each vertexviofHby a clusterAiand connect some pairs of vertices ofAiandAjif (vi,vj) is an edge of the graphH. As usual, we define the edge density betweenAiandAjasWe study the following problem. Given densities γijfor each edge (i,j) ∈E(H), one has to decide whether there exists a blow-up graphG[H], with edge densities at least γij, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic toH,i.e., noHappears as a transversal inG[H]. We calldcrit(H) the maximal value for which there exists a blow-up graphG[H] with edge densitiesd(Ai,Aj)=dcrit(H) ((vi,vj) ∈E(H)) not containingHin the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs.First, in the case of treeTwe give an efficient algorithm to decide whether a given set of edge densities ensures the existence of a transversalTin the blow-up graph. Then we give general bounds ondcrit(H) in terms of the maximal degree. In connection with the extremal structure, the so-called star decomposition is proved to give the best construction forH-transversal-free blow-up graphs for several graph classes. Our approach applies algebraic graph-theoretical, combinatorial and probabilistic tools.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

Inverting the Turán problem with chromatic number

2021 ◽  
Vol 344 (9) ◽  
pp. 112517
Author(s):  
Xiutao Zhu ◽  
Yaojun Chen
Keyword(s):  
Chromatic Number ◽  
Turan Problem ◽  
Turán Problem

scholarly journals On a Turán problem in weakly quasirandom 3-uniform hypergraphs

2018 ◽  
Vol 20 (5) ◽  
pp. 1139-1159 ◽  
Author(s):  
Christian Reiher ◽  
Vojtěch Rödl ◽  
Mathias Schacht
Keyword(s):  
Uniform Hypergraphs ◽  
Turan Problem ◽  
Turán Problem

scholarly journals An Ordered Turán Problem for Bipartite Graphs

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.

scholarly journals Unified approach to the generalized Turán problem and supersaturation

2022 ◽  
Vol 345 (3) ◽  
pp. 112743
Author(s):  
Dániel Gerbner ◽  
Zoltán Lóránt Nagy ◽  
Máté Vizer
Keyword(s):  
Unified Approach ◽  
Turan Problem ◽  
Turán Problem

Exact solution of the hypergraph Turán problem for k-uniform linear paths

COMBINATORICA ◽  
2014 ◽  
Vol 34 (3) ◽  
pp. 299-322 ◽  
Author(s):  
Zoltán Füredi ◽  
Tao Jiang ◽  
Robert Seiver
Keyword(s):  
Exact Solution ◽  
Turan Problem ◽  
Turán Problem

scholarly journals The Size of a Hypergraph and its Matching Number

2012 ◽  
Vol 21 (3) ◽  
pp. 442-450 ◽  
Author(s):  
HAO HUANG ◽  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Matching Number ◽  
Uniform Hypergraph ◽  
Edge Matching ◽  
Turan Problem ◽  
Turán Problem

More than forty years ago, Erdős conjectured that for any $t \leq \frac{n}{k}$, every k-uniform hypergraph on n vertices without t disjoint edges has at most max${\binom{kt-1}{k}, \binom{n}{k}-\binom{n-t+1}{k}\}$ edges. Although this appears to be a basic instance of the hypergraph Turán problem (with a t-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all $t < \frac{n}{3k^2}$. This improves upon the best previously known range $t = O\bigl(\frac{n}{k^3}\bigr)$, which dates back to the 1970s.

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