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.  

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.

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 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.

Parallel Multi-Cycle LES of an Optical Pent-Roof DISI Engine Under Motored Operating Conditions

2017 ◽  
Author(s):  
Noah Van Dam ◽  
Wei Zeng ◽  
Magnus Sjöberg ◽  
Sibendu Som
Keyword(s):  
Experimental Data ◽  
Direct Injection ◽  
Operating Conditions ◽  
New Strategy ◽  
Long Time ◽  
Order Of Magnitude ◽  
Structure Index ◽  
Number Of Cycles ◽  
The Many ◽  
Direct Injection Spark Ignition

The use of Large-eddy Simulations (LES) has increased due to their ability to resolve the turbulent fluctuations of engine flows and capture the resulting cycle-to-cycle variability. One drawback of LES, however, is the requirement to run multiple engine cycles to obtain the necessary cycle statistics for full validation. The standard method to obtain the cycles by running a single simulation through many engine cycles sequentially can take a long time to complete. Recently, a new strategy has been proposed by our research group to reduce the amount of time necessary to simulate the many engine cycles by running individual engine cycle simulations in parallel. With modern large computing systems this has the potential to reduce the amount of time necessary for a full set of simulated engine cycles to finish by up to an order of magnitude. In this paper, the Parallel Perturbation Methodology (PPM) is used to simulate up to 35 engine cycles of an optically accessible, pent-roof Direct-injection Spark-ignition (DISI) engine at two different motored engine operating conditions, one throttled and one un-throttled. Comparisons are made against corresponding sequential-cycle simulations to verify the similarity of results using either methodology. Mean results from the PPM approach are very similar to sequential-cycle results with less than 0.5% difference in pressure and a magnitude structure index (MSI) of 0.95. Differences in cycle-to-cycle variability (CCV) predictions are larger, but close to the statistical uncertainty in the measurement for the number of cycles simulated. PPM LES results were also compared against experimental data. Mean quantities such as pressure or mean velocities were typically matched to within 5–10%. Pressure CCVs were under-predicted, mostly due to the lack of any perturbations in the pressure boundary conditions between cycles. Velocity CCVs for the simulations had the same average magnitude as experiments, but the experimental data showed greater spatial variation in the root-mean-square (RMS). Conversely, circular standard deviation results showed greater repeatability of the flow directionality and swirl vortex positioning than the simulations.

Environmental Effects on Fatigue Crack Initiation in 2.25 Cr-1 Mo Steel and 316L Stainless Steel

1988 ◽  
Vol 110 (3) ◽  
pp. 240-246
Author(s):  
V. K. Mathews ◽  
T. S. Gross
Keyword(s):  
Stainless Steel ◽  
Fatigue Crack ◽  
Crack Initiation ◽  
Room Temperature ◽  
316L Stainless Steel ◽  
Silicone Oil ◽  
Fatigue Crack Initiation ◽  
Type A ◽  
Order Of Magnitude ◽  
Number Of Cycles

Blunt notch fatigue crack initiation tests for Type A387 2.25 Cr-1 Mo steel and 316L stainless steel were performed in air at room temperature, in silicone oil at room temperature, in V-131B coal process solvent at 100°C, and in chlorine-modified V-131B coal process solvent at 100°C. For both steels the most damaging environment was room temperature air. The number of cycles to initiate a crack were almost identical in the coal process solvent and the silicone oil for the Type A-387 steel. These two environments resulted in the longest crack initiation lifetime for the Type A-387 steel. The crack initiation lifetime for the Type A-387 steel in the chlorine modified V-131 B coal process solvent was roughly a factor of five less than the lifetime in the silicone oil and the unmodified coal process solvent. The crack initiation lifetime for the Type A-387 steel in room temperature air was a factor of 30 less than the lifetime in the silicone oil or the unmodified coal process solvent. The improvement of the crack initiation lifetime for the Type A-387 steel in the unmodified coal process solvent and the silicone oil is attributed to protection of the material from embrittlement from room temperature air. The decrease in crack initiation lifetime in the chlorine modified coal process solvent indicates that chlorine can be an active embrittling agent in the coal process solvent. The crack initiation lifetime for 316L stainless steel was longest in the silicone oil. The lifetime decreased somewhat in the unmodified coal process solvent with a further decrease for the chlorine modified coal solvent. The crack initiation lifetime in air was an order of magnitude lower than the lifetime in the silicone oil. The silicone oil and the coal process solvent apparently protected the 316L stainless from the embrittlement in air. However, the coal process solvent is not entirely inert as in the case of Type A-387 steel. The chlorine is an active embrittling agent for the 316L stainless steel in the coal process solvent.

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 $r \ge 3$.

Measuring the tensile strength of caked sugar produced from humidity cycling

2003 ◽  
Vol 217 (1) ◽  
pp. 41-47 ◽  
Author(s):  
M C Leaper ◽  
R J Berry ◽  
M S A Bradley ◽  
I Bridle ◽  
A R Reed ◽  
...  
Keyword(s):  
Tensile Strength ◽  
Relative Humidity ◽  
Temperature Cycling ◽  
A Value ◽  
Tensile Tester ◽  
Cent Relative Humidity ◽  
Order Of Magnitude ◽  
Number Of Cycles ◽  
The Relationship ◽  
Gas Bearing

The aim of this study was to develop moisture migration modelling in stored sugar due to the temperature cycling of the storage environment. This was done by forming cake of sugar in a gas-bearing tensile tester. The sugar and tester were enclosed in a controlled humidity system and the relative humidity was cycled between four hours at 70 per cent relative humidity and four hours at 20 per cent relative humidity. Liquid bridges formed between the sugar particles at high relative humidity, with the bridges hardening subsequently when the humidity was reduced. It was found that the relationship between the tensile strength and number of cycles could be approximated by the relationship σT = K √ N and the agglomerate tensile strength was in the order of 150Pa after 32 cycles. This suggests a soft cake rather than hard cakes with a tensile strength of twice this order of magnitude, which are formed, for example, in salt. A value of 1300kPa was obtained for the parameter representing the average strength of the bridge material in a simplified model of monosized spheres linked by pendular bridges in a system of uniform packing.

Graph Colouring with No Large Monochromatic Components

2008 ◽  
Vol 17 (4) ◽  
pp. 577-589 ◽  
Author(s):  
NATHAN LINIAL ◽  
JIŘÍ MATOUŠEK ◽  
OR SHEFFET ◽  
GÁBOR TARDOS
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
The Other ◽  
Graph Colouring ◽  
Connected Subgraph ◽  
Regular Graphs ◽  
Maximum Order ◽  
Asymptotically Optimal ◽  
Vertex Graph ◽  
Order Of Magnitude

For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let be any non-trivial minor-closed family of graphs. We show that mcc2(G) = O(n2/3) for any n-vertex graph G ∈ . This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such , and every fixed t we show that mcct(G)=O(n2/(t+1)). On the other hand, we have examples of graphs G with no Kt+3 minor and with mcct(G)=Ω(n2/(2t−1)).It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos proved mcc2(G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc2(G)=Ω(n), and more sharply, for every ϵ > 0 there exists cϵ > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ϵ for all subgraphs, and with mcc2(G) ≥ cϵn. For 6-regular graphs it is known only that the maximum order of magnitude of mcc2 is between $\sqrt n$ and n.We also offer a Ramsey-theoretic perspective of the quantity mcct(G).

scholarly journals Repetition Number of Graphs

10.37236/96 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yair Caro ◽  
Douglas B. West
Keyword(s):  
Minimum Degree ◽  
Line Graph ◽  
Outerplanar Graph ◽  
Line Graphs ◽  
Outerplanar Graphs ◽  
Free Graph ◽  
Maximal Outerplanar Graphs ◽  
Vertex Graph ◽  
Repetition Number ◽  
Maximum Multiplicity

Every $n$-vertex graph has two vertices with the same degree (if $n\ge2$). In general, let rep$(G)$ be the maximum multiplicity of a vertex degree in $G$. An easy counting argument yields rep$(G)\ge n/(2d-2s+1)$, where $d$ is the average degree and $s$ is the minimum degree of $G$. Equality can hold when $2d$ is an integer, and the bound is approximately sharp in general, even when $G$ is restricted to be a tree, maximal outerplanar graph, planar triangulation, or claw-free graph. Among large claw-free graphs, repetition number $2$ is achievable, but if $G$ is an $n$-vertex line graph, then rep$(G)\ge{1\over4}n^{1/3}$. Among line graphs of trees, the minimum repetition number is $\Theta(n^{1/2})$. For line graphs of maximal outerplanar graphs, trees with perfect matchings, or triangulations with 2-factors, the lower bound is linear.

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