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)].

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 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 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 $r \ge 3$.

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.

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 Another Infinite Sequence of Dense Triangle-Free Graphs

10.37236/1381 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Stephan Brandt ◽  
Tomaž Pisanski
Keyword(s):  
Structural Properties ◽  
Chromatic Number ◽  
Infinite Sequence ◽  
Minimum Degree ◽  
Free Graph ◽  
The Core ◽  
Graph Homomorphisms ◽  
Natural Way

The core is the unique homorphically minimal subgraph of a graph. A triangle-free graph with minimum degree $\delta > n/3$ is called dense. It was observed by many authors that dense triangle-free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs in this sequence are 3-colourable. Only two additional cores with chromatic number 4 were known. We show that the additional graphs are the initial terms of a second infinite sequence of cores.

scholarly journals New Results on Degree Sequences of Uniform Hypergraphs

10.37236/3414 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Sarah Behrens ◽  
Catherine Erbes ◽  
Michael Ferrara ◽  
Stephen G. Hartke ◽  
Benjamin Reiniger ◽  
...  
Keyword(s):  
Efficient Algorithm ◽  
Necessary Conditions ◽  
Degree Sequence ◽  
Sufficient Conditions ◽  
Degree Sequences ◽  
Uniform Hypergraph ◽  
Graphic Sequence ◽  
Uniform Hypergraphs ◽  
Open Question

A sequence of nonnegative integers is $k$-graphic if it is the degree sequence of a $k$-uniform hypergraph. The only known characterization of $k$-graphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary conditions appear in the literature, there are few conditions that imply a sequence is $k$-graphic. In light of this, we present sharp sufficient conditions for $k$-graphicality based on a sequence's length and degree sum.Kocay and Li gave a family of edge exchanges (an extension of 2-switches) that could be used to transform one realization of a 3-graphic sequence into any other realization. We extend their result to $k$-graphic sequences for all $k \geq 3$. Finally we give several applications of edge exchanges in hypergraphs, including generalizing a result of Busch et al. on packing graphic sequences.

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

Kinetics of Fertilization in the Sea Urchin: A Comparison of Methods

1970 ◽  
Vol 52 (2) ◽  
pp. 455-468
Author(s):  
R. PRESLEY ◽  
P. F. BAKER
Keyword(s):  
Sea Urchin ◽  
Uranyl Nitrate ◽  
Sea Water ◽  
Fertilization Rate ◽  
The Other ◽  
Jelly Coat ◽  
Cortical Responses ◽  
Direct Counting ◽  
Further Development ◽  
Kinetics Of

1. A method is described for the direct counting of male pronuclei in recently fertilized sea-urchin eggs. 2. Using this method, fertilization rate determinations were made to compare 30% artificial sea water (A.S.W.), isotonic KCl, sea water containing lauryl sulphate, calcium-free and magnesium-free A.S.W. containing EDTA, and sea water containing uranyl nitrate, as agents blocking fertilization but permitting further development of previously fertilized eggs. 3. 30% A.S.W. was found to be less satisfactory than the other agents, lacking instant effect, and tending to promote polyspermy. The other agents all gave sigmoid rate curves, that of uranyl nitrate lagging 15-25 sec. behind the others. 4. Evidence was found that uranyl nitrate acts at a later stage in fertilization than the other agents. 5. Sigmoid rate curves were found, except with 30% A.S.W., when eggs with the bulk of the jelly coat removed, and nicotine-treated eggs, were fertilized. 6. Analysis of sperm distribution among eggs from samples fertilized for more than 40 sec. confirmed that re-fertilization takes place at a lower rate than primary fertilization. 7. The processes blocked by KCl and uranyl nitrate were found to precede the cortical responses to fertilization, and the termination of nicotine sensitivity.

scholarly journals An extension of Waring's problems

1933 ◽  
Vol 232 (707-720) ◽  
pp. 1-26 ◽  
Keyword(s):  
The Other ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Large N ◽  
Hardy Littlewood Method

An interesting extension of Waring’s famous problem is the following:— Can every sufficiently large n be expressed, as the sum of s almost equal k-th powers; or, more generally, can every sufficiently large n be expressed as the sum of s positive k-th powers almost proportional to s arbitrarily assigned positive numbers Xl5 x2, ... x, ? I have developed two methods to discuss this problem, one based on the Hardy-Littlewood method for the solution of WARING’s problem and the other on the new VINOGRADOFF method* for the solution of the same problem. In this paper I shall discuss the case k3 by the first of these methods. The case of five or more squares may be treated in the same way, and the results are similar; somewhat deeper and more troublesome analysis is required to deal with the case of four squares. The principle of the method employed here is that of weighting ” the various representations of n as the sum of s k-th powers in such a way as to make predominant the particular representation of which we are in search.

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