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 Lower Bounds for the Size of Random Maximal $H$-Free Graphs

10.37236/93 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Guy Wolfovitz
Keyword(s):  
Lower Bound ◽  
Random Process ◽  
Lower Bounds ◽  
Random Permutation ◽  
Bipartite Graphs ◽  
Expected Number ◽  
Free Graph ◽  
Vertex Graph

We consider the next random process for generating a maximal $H$-free graph: Given a fixed graph $H$ and an integer $n$, start by taking a uniformly random permutation of the edges of the complete $n$-vertex graph $K_n$. Then, traverse the edges of $K_n$ according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving $n$-vertex graph - unless its addition creates a copy of $H$. The result of this process is a maximal $H$-free graph ${\Bbb M}_n(H)$. Our main result is a new lower bound on the expected number of edges in ${\Bbb M}_n(H)$, for $H$ that is regular, strictly $2$-balanced. As a corollary, we obtain new lower bounds for Turán numbers of complete, balanced bipartite graphs. Namely, for fixed $r \ge 5$, we show that ex$(n, K_{r,r}) = \Omega(n^{2-2/(r+1)}(\ln\ln n)^{1/(r^2-1)})$. This improves an old lower bound of Erdős and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in ${\Bbb M}_n(H)$, conditioned on the event that the edge is traversed relatively (but not trivially) early during the process.

scholarly journals Paths of Length Three are $K_{r+1}$-Turán-Good

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Kyle Murphy ◽  
JD Nir
Keyword(s):  
Free Graph ◽  
Path Graph ◽  
Flag Algebras ◽  
Turan Problem ◽  
Turán Problem ◽  
Good For ◽  
Turán Graph

The generalized Turán problem ex$(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.

scholarly journals Some Exact Results and New Asymptotics for Hypergraph Turán Numbers

2002 ◽  
Vol 11 (3) ◽  
pp. 299-309 ◽  
Author(s):  
DHRUV MUBAYI
Keyword(s):  
Lower Bound ◽  
Explicit Construction ◽  
Bipartite Graphs ◽  
Probabilistic Methods ◽  
Exact Results ◽  
Open Problems ◽  
Asymptotic Value ◽  
The Family ◽  
Turán Number

Given a family [Fscr ] of r-graphs, let ex(n, [Fscr ]) be the maximum number of edges in an n-vertex r-graph containing no member of [Fscr ]. Let C(r)4 denote the family of r-graphs with distinct edges A, B, C, D, such that A ∩ B = C ∩ D = Ø and A ∪ B = C ∪ D. For s1 [les ] … [les ] sr, let K(r) (s1,…,sr) be the complete r-partite r-graph with parts of sizes s1,…,sr.Füredi conjectured over 15 years ago that ex(n,C(3)4) [les ] (n2) for sufficiently large n. We prove the weaker resultex(n, {C(3)4, K(3)(1,2,4)}) [les ] (n2).Generalizing a well-known conjecture for the Turán number of bipartite graphs, we conjecture thatex(n, K(r)(s1,…,sr)) = Θ(nr−1/s),where s = Πr−1i=1si. We prove this conjecture when s1 = … = sr−2 = 1 and(i) sr−1 = 2, (ii) sr−1 = sr = 3, (iii)sr > (sr−1−1)!.In cases (i) and (ii), we determine the asymptotic value of ex(n,K(r)(s1,…,sr)).We also provide an explicit construction givingex(n,K(3)(2,2,3)) > (1/6−o(1))n8/3.This improves upon the previous best lower bound of Ω(n29/11) obtained by probabilistic methods. Several related open problems are also presented.

scholarly journals Degree Sequences of $F$-Free Graphs

10.37236/1966 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Oleg Pikhurko ◽  
Anusch Taraz
Keyword(s):  
Chromatic Number ◽  
Degree Sequence ◽  
The Other ◽  
Degree Sequences ◽  
Regularity Lemma ◽  
Free Graph ◽  
Large N ◽  
Turan Problem ◽  
Turán Problem ◽  
Direct Counting

Let $F$ be a fixed graph of chromatic number $r+1$. We prove that for all large $n$ the degree sequence of any $F$-free graph of order $n$ is, in a sense, close to being dominated by the degree sequence of some $r$-partite graph. We present two different proofs: one goes via the Regularity Lemma and the other uses a more direct counting argument. Although the latter proof is longer, it gives better estimates and allows $F$ to grow with $n$. As an application of our theorem, we present new results on the generalization of the Turán problem introduced by Caro and Yuster [Electronic J. Combin. 7 (2000)].

scholarly journals Turán Number of an Induced Complete Bipartite Graph Plus an Odd Cycle

2018 ◽  
Vol 28 (2) ◽  
pp. 241-252 ◽  
Author(s):  
BEKA ERGEMLIDZE ◽  
ERVIN GYŐRI ◽  
ABHISHEK METHUKU
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Free Graph ◽  
Odd Cycle ◽  
Turán Number

Let k ⩾ 2 be an integer. We show that if s = 2 and t ⩾ 2, or s = t = 3, then the maximum possible number of edges in a C2k+1-free graph containing no induced copy of Ks,t is asymptotically equal to (t − s + 1)1/s(n/2)2−1/s except when k = s = t = 2.This strengthens a result of Allen, Keevash, Sudakov and Verstraëte [1], and answers a question of Loh, Tait, Timmons and Zhou [14].

scholarly journals A Multipartite Version of the Turán Problem – Density Conditions and Eigenvalues

10.37236/533 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Zoltán Lóránt Nagy
Keyword(s):  
Edge Density ◽  
Forbidden Subgraphs ◽  
Arbitrary Graph ◽  
Turan Problem ◽  
Strongly Connected ◽  
Critical Edge ◽  
Minimum Number ◽  
Turán Problem ◽  
Extremal Structure

In this paper we propose a multipartite version of the classical Turán problem of determining the minimum number of edges needed for an arbitrary graph to contain a given subgraph. As it turns out, here the non-trivial problem is the determination of the minimal edge density between two classes that implies the existence of a given subgraph. We determine the critical edge density for trees and cycles as forbidden subgraphs, and give the extremal structure. Surprisingly, this critical edge density is strongly connected to the maximal eigenvalue of the graph. Furthermore, we give a sharp upper and lower bound in general, in terms of the maximum degree of the forbidden graph.

scholarly journals Generalized Planar Turán Numbers

10.37236/9603 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ervin Győri ◽  
Addisu Paulos ◽  
Nika Salia ◽  
Casey Tompkins ◽  
Oscar Zamora
Keyword(s):  
Planar Graph ◽  
Exact Result ◽  
Free Graph ◽  
Vertex Graph ◽  
Turan Problems ◽  
Order Of Magnitude ◽  
Turán Problem ◽  
Host Graph ◽  
Number Of Cycles ◽  
Fixed Tree

In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.  

Role of the family focused technologies in the clinical course of pregnancy at women of high obstetric risk

2016 ◽  
pp. 64-66
Author(s):  
S.Yu. Vdovichenko ◽  
Keyword(s):  
Comparison Group ◽  
Traditional Approach ◽  
Married Couple ◽  
Premature Births ◽  
Individual Support ◽  
The Family ◽  
Risk Patients ◽  
Individual Preparation

The objective: to show a role of the family focused technologies in depression of frequency of pathology of pregnancy at women of high obstetric risk. Patients and methods. For determination of efficiency of prophylaxis of pathology of pregnancy on the basis of use of the family focused technologies complex clinical-psychological and laboratory and tool examination of 300 women with factors of obstetric risk which were divided into two groups was conducted. In the main group – 182 women with motivation on partner labors to which provided training on system of individual preparation of married couple to labors. The comparison group consisted of 118 women who were not in prenatal training and had individual support in childbirth, with the traditional approach to pain management. Results. Use of the family focused technologies during pregnancy allows to reduce significantly the frequency of the main complications of pregnancy, especially not incubation and premature births. Conclusion. In our opinion, the technique is simple, available and can widely be used in practical health care at women with high obstetric risk. Key words: obstetric risk, the family focused technologies, prophylaxis.

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.

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