A Generalized Turán Problem and its Applications

2018 ◽  
Vol 2020 (11) ◽  
pp. 3417-3452 ◽  
Author(s):  
Lior Gishboliner ◽  
Asaf Shapira
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Polynomial Function ◽  
Upper And Lower Bounds ◽  
Turan Problem ◽  
Turán Problem ◽  
Polynomial Bounds ◽  
Removal Lemma ◽  
Special Case

Abstract The investigation of conditions guaranteeing the appearance of cycles of certain lengths is one of the most well-studied topics in graph theory. In this paper we consider a problem of this type that asks, for fixed integers ℓ and k, how many copies of the k-cycle guarantee the appearance of an ℓ-cycle? Extending previous results of Bollobás–Gy̋ri–Li and Alon–Shikhelman, we fully resolve this problem by giving tight (or nearly tight) bounds for all values of ℓ and k. We also present a somewhat surprising application of the above mentioned estimates to the study of the graph removal lemma. Prior to this work, all bounds for removal lemmas were either polynomial or there was a tower-type gap between the best-known upper and lower bounds. We fill this gap by showing that for every super-polynomial function $f(\varepsilon )$, there is a family of graphs ${\mathcal F}$, such that the bounds for the ${\mathcal F}$ removal lemma are precisely given by $f(\varepsilon )$. We thus obtain the 1st examples of removal lemmas with tight super-polynomial bounds. A special case of this result resolves a problem of Alon and the 2nd author, while another special case partially resolves a problem of Goldreich.

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

scholarly journals On the number of excursion sets of planar Gaussian fields

2020 ◽  
Vol 178 (3-4) ◽  
pp. 655-698
Author(s):  
Dmitry Beliaev ◽  
Michael McAuley ◽  
Stephen Muirhead
Keyword(s):  
Plane Wave ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Gaussian Fields ◽  
Important Special Case ◽  
Level Densities ◽  
Excursion Sets ◽  
Nodal Set ◽  
Different Types ◽  
Special Case

Abstract The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

SIMULATIONS OF UNARY ONE-WAY MULTI-HEAD FINITE AUTOMATA

2014 ◽  
Vol 25 (07) ◽  
pp. 877-896 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER ◽  
MATTHIAS WENDLANDT
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Unary Languages ◽  
Order Of Magnitude ◽  
Simulation Results ◽  
Special Case ◽  
Nondeterministic Finite Automaton

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case upper and lower bounds that are tight in the order of magnitude are shown. For the latter case we obtain an upper bound of O(n2k) and a lower bound of Ω(nk) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. In addition, we study how the conversion costs vary in the special case of finite and, in particular, of singleton unary lanuages. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

scholarly journals Simplified group activity selection with group size constraints

2021 ◽  
Author(s):  
Andreas Darmann ◽  
Janosch Döcker ◽  
Britta Dorn ◽  
Sebastian Schneckenburger
Keyword(s):  
Computational Complexity ◽  
Simple Model ◽  
Lower Bounds ◽  
Group Activity ◽  
Core Stability ◽  
Upper And Lower Bounds ◽  
Size Constraints ◽  
Broad Variety ◽  
Special Case ◽  
Additional Constraints

AbstractSeveral real-world situations can be represented in terms of agents that have preferences over activities in which they may participate. Often, the agents can take part in at most one activity (for instance, since these take place simultaneously), and there are additional constraints on the number of agents that can participate in an activity. In such a setting, we consider the task of assigning agents to activities in a reasonable way. We introduce the simplified group activity selection problem providing a general yet simple model for a broad variety of settings, and start investigating its special case where upper and lower bounds of the groups have to be taken into account. We apply different solution concepts such as envy-freeness and core stability to our setting and provide a computational complexity study for the problem of finding such solutions.

Quantum uncertainty relations of Tsallis relative $\alpha$ entropy coherence baed on MUBs

2021 ◽  
Author(s):  
ZHANG Fu Gang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Mutually Unbiased Bases ◽  
Uncertainty Relations ◽  
Quantum Uncertainty ◽  
Single Qubit ◽  
Qubit System ◽  
Special Case

Abstract In this paper, we discuss quantum uncertainty relations of Tsallis relative $\alpha$ entropy coherence for a single qubit system based on three mutually unbiased bases. For $\alpha\in[\frac{1}{2},1)\cup(1,2]$, the upper and lower bounds of sums of coherence are obtained. However, the above results cannot be verified directly for any $\alpha\in(0,\frac{1}{2})$. Hence, we only consider the special case of $\alpha=\frac{1}{n+1}$, where $n$ is a positive integer, and we obtain the upper and lower bounds. By comparing the upper and lower bounds, we find that the upper bound is equal to the lower bound for the special $\alpha=\frac{1}{2}$, and the differences between the upper and the lower bounds will increase as $\alpha$ increases. Furthermore, we discuss the tendency of the sum of coherence, and find that it has the same tendency with respect to the different $\theta$ or $\varphi$, which is opposite to the uncertainty relations based on the R\'{e}nyi entropy and Tsallis entropy.

New and improved results on the signed (total) k-domination number of graphs

2017 ◽  
Vol 09 (02) ◽  
pp. 1750024 ◽  
Author(s):  
D. A. Mojdeh ◽  
Babak Samadi
Keyword(s):  
Graph Theory ◽  
Extremal Problem ◽  
Lower Bounds ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Free Graph ◽  
Uniform Approach ◽  
Number Formula ◽  
Graph Parameters ◽  
Two Parameters

In this paper, we study the signed [Formula: see text]-domination and its total version in graphs. By a simple uniform approach we give some new upper and lower bounds on these two parameters of a graph in terms of several different graph parameters. In this way, we can improve and generalize some results in literature. Moreover, we make use of the well-known theorem of Turán [On an extremal problem in graph theory, Math. Fiz. Lapok 48 (1941) 436–452] to bound the signed total [Formula: see text]-domination number, [Formula: see text], of a [Formula: see text]-free graph [Formula: see text] for [Formula: see text].

scholarly journals The treewidth of 2-section of hypergraphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Ke Liu ◽  
Mei Lu
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Upper And Lower Bounds ◽  
Average Rank ◽  
Linear Hypergraph ◽  
Algorithmic Graph Theory

Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.

SOME NEW RELIABILITY BOUNDS FOR SUMS OF NBUE RANDOM VARIABLES

2010 ◽  
Vol 25 (1) ◽  
pp. 83-102
Author(s):  
Steven G. From
Keyword(s):  
Lower Bounds ◽  
Random Variables ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Important Special Case ◽  
Lifetime Distributions ◽  
Survivor Function ◽  
Numerical Studies ◽  
Special Case ◽  
Better Than

In this article, we discuss some new upper and lower bounds for the survivor function of the sum of n independent random variables each of which has an NBUE (new better than used in expectation) distribution. In some cases, only the means of the random variables are assumed known. These bounds are compared to the sharp bounds given in Cheng and Lam [6], which requires both means and variances known. Although the new bounds are not sharp, they often produce better upper bounds for the survivor function in the extreme right tail of many NBUE lifetime distributions, an important special case in applications. Moreover, a lower bound exists in one case not handled by the lower bounds of Theorem 3 in Cheng and Lam [6]. Numerical studies are presented along with theoretical discussions.

Microcomputer-Assisted Mathematics: Using the Microcomputer to Find the Zeros of a Polynomial Function

1986 ◽  
Vol 79 (7) ◽  
pp. 554-558
Author(s):  
Charles D. Friesen
Keyword(s):  
Lower Bounds ◽  
Mathematics Teachers ◽  
Polynomial Function ◽  
Upper And Lower Bounds ◽  
Polynomial Functions

Mathematics teachers are finding more and more opportunities to use the microcomputer as a tool in their classes. The synthetic-division program listed in the Appendix has been used in advanced algebra, analysis, and calculus classes. This program can be used in a variety of settings. This article will deal with its use in searching for the zeros of polynomial functions. The program can also be used to evaluate polynomials, obtain quotients, and search for the upper and lower bounds of the zeros of a polynomial.

Connectivity Coding: New Perspectives for Mesh Compression (Connectivity Coding: Neue Perspektiven für die Kompression von Dreiecksnetzen)

2002 ◽  
Vol 44 (6) ◽  
Author(s):  
Stefan Gumhold
Keyword(s):  
Graph Theory ◽  
Computer Graphics ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Mesh Compression ◽  
Triangle Meshes ◽  
Connectivity Graph ◽  
Planar Triangulation ◽  
Planar Regions

Compact encodings of the connectivity of planar triangulations is a very important subject not only in graph theory but also in computer graphics. For triangle meshes used in computer graphics the topologically planar regions dominate by far. New results by Isenburg et al. [6] even show that the connectivity is sufficient to describe shape by itself. Most coding methods for planar triangulations can be extended to manifolds of bounded genus with the same upper and lower bounds on the bit rate.In 1962 Tutte enumerated the number of different planar triangulations and his results show, that at least 3.245 bits per vertex are necessary to encode the connectivity graph of planar triangulations. In this article we improve the so far best upper bound [5] and show that the connectivity of a planar triangulation can be encoded with less than 3.525 bits per vertex, while ensuring a linear run time for encoding and decoding.

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