scholarly journals Rainbow Turán Problems for Paths and Forests of Stars

10.37236/6430 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Daniel Johnston ◽  
Cory Palmer ◽  
Amites Sarkar
Keyword(s):  
Systematic Study ◽  
Colored Graph ◽  
Turán Number ◽  
Turan Problems ◽  
Different Color

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the rainbow Turán number of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte [Combinatorics, Probability and Computing 16 (2007)]. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $l$ edges, disproving a conjecture in the aforementioned paper for $l=4$.

scholarly journals On zero-sum turan problems of Bialostocki and Dierker

1992 ◽  
Vol 53 (3) ◽  
pp. 402-407 ◽  
Author(s):  
Y. Caro ◽  
Y. Roditty
Keyword(s):  
Additive Group ◽  
The Family ◽  
Turán Number ◽  
Turan Problems ◽  
Zero Sum

AbstractAssume G is a graph with m edges. By T(n, G) we denote the classical Turan number, namely, the maximum possible number of edges in a graph H on n vertices without a copy of G. Similarly if G is a family of graphs then H does not have a copy of any member of the family. A Zk-colouring of a graph G is a colouring of the edges of G by Zk, the additive group of integers modulo k, avoiding a copy of a given graph H, for which the sum of the values on its edges is 0 (mod k). By the Zero-Sum Turan number, denoted T(n, G, Zk), k¦m, we mean the maximum number of edges in a Zk-colouring of a graph on n vertices that contains no zero-sum (mod k) copy of G. Here we mainly solve two problems of Bialostocki and Dierker [6].Problem 1. Determine T(n, tK2, Zk) for ¦|t. In particular, is it true that T(n, tK2, Zk) = T(n, (t+k-1)K2)?Problem 2. Does there exist a constant c(t, k) such that T(n, Ft, Zk) ≦ c(t, k)n, where Ft is the family of cycles of length at least t?

scholarly journals Turán Density of $2$-Edge-Colored Bipartite Graphs with Application on $\{2, 3\}$-Hypergraphs

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Shuliang Bai ◽  
Linyuan Lu
Keyword(s):  
Bipartite Graphs ◽  
Important Application ◽  
Edge Density ◽  
Colored Graph ◽  
Colored Graphs ◽  
Turan Problems ◽  
Vertex Set

We consider the Turán problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ where $E_r$ and $E_b$ do not have to be disjoint. The Turán density $\pi(H)$ of $H$ is defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/\binom{n}{2}$ is the formula to measure the edge density of $G_n$. We will determine the Turán densities of all $2$-edge-colored bipartite graphs. We also give an important application on the Turán problems of $\{2, 3\}$-hypergraphs.

scholarly journals On the Rainbow Turán number of paths

10.37236/7889 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Beka Ergemlidze ◽  
Ervin Győri ◽  
Abhishek Methuku
Keyword(s):  
Systematic Study ◽  
Edge Coloring ◽  
Turán Number ◽  
Proper Edge Coloring

Let $F$ be a fixed graph. The rainbow Turán number of $F$ is defined as the maximum number of edges in a graph on $n$ vertices that has a proper edge-coloring with no rainbow copy of $F$ (i.e., a copy of $F$ all of whose edges have different colours). The systematic study of such problems was initiated by Keevash, Mubayi, Sudakov and Verstraëte.  In this paper, we show that the rainbow Turán number of a path with $k+1$ edges is less than $\left(9k/7+2\right) n$, improving an earlier estimate of Johnston,  Palmer and Sarkar.

scholarly journals A steady-state grouping genetic algorithm for the rainbow spanning tree problem

2021 ◽  
Author(s):  
Sudishna Ghoshal ◽  
shyam sundar
Keyword(s):  
Genetic Algorithm ◽  
Steady State ◽  
Spanning Tree ◽  
Colored Graph ◽  
Solution Quality ◽  
Grouping Genetic Algorithm ◽  
Minimum Number ◽  
Grouping Problem ◽  
Different Color ◽  
Acyclic Subgraph

Abstract Given a connected, undirected and edge-colored graph, the rainbow spanning tree (RSF) problem aims to find a rainbow spanning forest with the minimum number of rainbow trees, where a rainbow tree is a connected acyclic subgraph of the graph whose each edge is associated with a different color. This problem is $NP$-Hard and finds several applications in distinguishing among various types of connections. Being a grouping problem, this paper proposes a steady-state grouping genetic algorithm (SSGGA) for the RSF problem. To the best of our knowledge, this is the first work on steady-state grouping genetic algorithm for this problem. While keeping in view of grouping aspects of the problem, each individual, in the proposed SSGGA, is encoded as a group of rainbow trees, and accordingly a problem-specific crossover operator is designed. Moreover, SSGGA uses the idea of two steps in its replacement scheme. All such elements of SSGGA coordinate effectively and overall help in finding high quality solutions. Computational results obtained over a set of benchmark instances show that overall SSGGA, in terms of solution quality, is superior to all other existing approaches in the literature for this problem.

Quantitative parallel EELS spectrum imaging developed as a new tool in AEM

1992 ◽  
Vol 50 (2) ◽  
pp. 1202-1203
Author(s):  
Gianluigi Botton ◽  
Gilles L'espérance
Keyword(s):  
Statistical Analysis ◽  
Spatial Resolution ◽  
Personal Computer ◽  
Systematic Study ◽  
Present Author ◽  
Chemical Elements ◽  
Detection Limits ◽  
Research Groups ◽  
Quantitative Performance ◽  
Spectrum Imaging

As interest for parallel EELS spectrum imaging grows in laboratories equipped with commercial spectrometers, different approaches were used in recent years by a few research groups in the development of the technique of spectrum imaging as reported in the literature. Either by controlling, with a personal computer both the microsope and the spectrometer or using more powerful workstations interfaced to conventional multichannel analysers with commercially available programs to control the microscope and the spectrometer, spectrum images can now be obtained. Work on the limits of the technique, in terms of the quantitative performance was reported, however, by the present author where a systematic study of artifacts detection limits, statistical errors as a function of desired spatial resolution and range of chemical elements to be studied in a map was carried out The aim of the present paper is to show an application of quantitative parallel EELS spectrum imaging where statistical analysis is performed at each pixel and interpretation is carried out using criteria established from the statistical analysis and variations in composition are analyzed with the help of information retreived from t/γ maps so that artifacts are avoided.

A SYSTEMATIC STUDY OF 2s2 2pk - 2s2pk+l - 2pk+2 TRANSITIONS IN MULTIPLY-CHARGED Cl IONS.

1979 ◽  
Vol 40 (C1) ◽  
pp. C1-208-C1-210 ◽  
Author(s):  
J. P. Forester ◽  
D. J. Pegg ◽  
P. M. Griffin ◽  
G. D. Alton ◽  
S. B. Elston ◽  
...  
Keyword(s):  
Systematic Study ◽  
Multiply Charged

The solubility of calcium carbonate  in green liquor handling systems

TAPPI Journal ◽  
2019 ◽  
Vol 18 (10) ◽  
pp. 595-602
Author(s):  
ALISHA GIGLIO ◽  
VLADIMIROS G. PAPANGELAKIS ◽  
HONGHI TRAN
Keyword(s):  
Calcium Carbonate ◽  
Systematic Study ◽  
Kraft Pulp ◽  
Thermodynamic Simulation ◽  
Solubility Data ◽  
Pulp Mills ◽  
Persistent Problem ◽  
Green Liquor ◽  
Kraft Pulp Mills ◽  
Salt Systems

The formation of hard calcite (CaCO3) scale in green liquor handling systems is a persistent problem in many kraft pulp mills. CaCO3 precipitates when its concentration in the green liquor exceeds its solubility. While the solubility of CaCO3 in water is well known, it is not so in the highly alkaline green liquor environment. A systematic study was conducted to determine the solubility of CaCO3 in green liquor as a function of temperature, total titratable alkali (TTA), causticity, and sulfidity. The results show that the solubility increases with increased temperature, increased TTA, decreased causticity, and decreased sulfidity. The new solubility data was incorporated into OLI (a thermodynamic simulation program for aqueous salt systems) to generate a series of CaCO3 solubility curves for various green liquor conditions. The results help explain how calcite scale forms in green liquor handling systems.

Sign in / Sign up

Export Citation Format

Share Document