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.

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.

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.

Determination of the minimum number of technological bases of a part

2020 ◽  
pp. 73-75
Author(s):  
B.M. Bazrov ◽  
T.M. Gaynutdinov
Keyword(s):  
Cost Price ◽  
Labor Input ◽  
Part Surface ◽  
Technological Base ◽  
Size Accuracy ◽  
Minimum Number ◽  
The Cost ◽  
Input Cost ◽  
Selection Of

The selection of technological bases is considered before the choice of the type of billet and the development of the route of the technological process. A technique is proposed for selecting the minimum number of sets of technological bases according to the criterion of equality in the cost price of manufacturing the part according to the principle of unity and combination of bases at this stage. Keywords: part, surface, coordinating size, accuracy, design and technological base, labor input, cost price. [email protected]

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 Determination of the minimum number of lymph nodes to examine to maximize survival in patients with esophageal carcinoma: Data from the Surveillance Epidemiology and End Results database

2010 ◽  
Vol 139 (3) ◽  
pp. 612-620 ◽  
Author(s):  
Shawn S. Groth ◽  
Beth A. Virnig ◽  
Bryan A. Whitson ◽  
Todd E. DeFor ◽  
Zhong-ze Li ◽  
...  
Keyword(s):  
Lymph Nodes ◽  
Esophageal Carcinoma ◽  
Minimum Number ◽  
End Results

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

A Hybrid Approach to Computer-Aided Process Planning for Prismatic Parts

1994 ◽  
Author(s):  
Y. F. Zhang ◽  
A. Y. C. Nee ◽  
J. Y. H. Fuh
Keyword(s):  
Process Planning ◽  
Hybrid Approach ◽  
Optimization Method ◽  
Optimal Operation ◽  
Operation Sequencing ◽  
Computer Aided Process Planning ◽  
Prismatic Parts ◽  
Minimum Number ◽  
Set Up

Abstract One of the most difficult tasks in automated process planning is the determination of operation sequencing. This paper describes a hybrid approach for identifying the optimal operation sequence of machining prismatic parts on a three-axis milling machining centre. In the proposed methodology, the operation sequencing is carried out in two levels of planning: set-up planning and operation planning. Various constraints on the precedence relationships between features are identified and rules and heuristics are created. Based on the precedence relationships between features, an optimization method is developed to find the optimal plan(s) with minimum number of set-ups in which the conflict between the feature precedence relationships and set-up sequence is avoided. For each set-up, an optimal feature machining sequence with minimum number of tool changes is also determined using a developed algorithm. The proposed system is still under development and the hybrid approach is partially implemented. An example is provided to demonstrate this approach.

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

A Simple Testing Procedure for near Infrared Instruments

1998 ◽  
Vol 6 (A) ◽  
pp. A163-A170 ◽  
Author(s):  
B. Barabás
Keyword(s):  
Error Compensation ◽  
Near Infrared ◽  
Testing Procedure ◽  
Wheat Protein ◽  
Calibration Equation ◽  
Minimum Number ◽  
Mean Square Difference ◽  
Infrared Instruments ◽  
Protein Measurement

The testing and adjusting procedure of near infrared (NIR) spectrophotometers is based on the measurement of some standards and, if necessary, on the adjustment of the constants in the calibration equation. For this work some use few standards, whereas others use 20 or more. This work was aimed to determine the range of error compensation and the minimum number of standards required. The experiments were applied to wheat protein measurement using two scanning spectrophotometers. The errors in the NIR measurements were characterised as bias, skew, error derived from skew ( Eskew) and standard error of difference corrected for bias and skew ( SEDc) parameters and supposed that errors derived from the change in the wavelength or reflectance of the instrument. The confidence intervals of bias and skew, derived from duplicate measurements of various numbers of wheat standards, were used to determine the minimum number of standards required. The range of error compensation was defined with those bias values, where SEDc was smaller, than an acceptable limit. The range of compensation corresponded to a bias value of ± 8 g kg−1 for wheat protein measurements. The detection of error of measurements required 4 wheat standards. The elimination of errors of bias and skew required 9 standards within the above limits. The developed procedure was tested in case of real instrument error. Diminishing a bias from 5.2 g kg−1 to 0.7 g kg−1 and the root mean square difference ( RMSD) to an acceptable level required the use of 9 standards, similar to the model experiment. The simplicity and rapidity (about 10 min) of the procedure enabled the routine test of NIR instruments. The range of error compensation and the number of standards referred to wheat protein. The simple modelling procedure proved also suitable for the determination of these values for other components and under other measuring conditions.

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
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