Perfect Graphs of Fixed Density: Counting and Homogeneous Sets

2012 ◽  
Vol 21 (5) ◽  
pp. 661-682 ◽  
Author(s):  
JULIA BÖTTCHER ◽  
ANUSCH TARAZ ◽  
ANDREAS WÜRFL
Keyword(s):  
Entropy Function ◽  
Perfect Graphs ◽  
Linear Size ◽  
Image Position ◽  
Almost All ◽  
Binary Entropy

For c ∈ (0,1) let n(c) denote the set of n-vertex perfect graphs with density c and let n(c) denote the set of n-vertex graphs without induced C5 and with density c.We show that with otherwise, where H is the binary entropy function.Further, we use this result to deduce that almost all graphs in n(c) have homogeneous sets of linear size. This answers a question raised by Loebl and co-workers.

Almost all Berge Graphs are Perfect

1992 ◽  
Vol 1 (1) ◽  
pp. 53-79 ◽  
Author(s):  
Hans Jürgen Prömel ◽  
Angelika Steger
Keyword(s):  
Perfect Graphs ◽  
Perfect Graph ◽  
Berge Graphs ◽  
Image Position ◽  
Almost All

Let Per f(n) denote the set of all perfect graphs on n vertices and let Berge(n) denote the set of all Berge graphs on n vertices. The strong perfect graph conjecture states that Per f(n) = Berge(n) for all n. In this paper we prove that this conjecture is at least asymptotically true, i.e. we show that

Perfectly orderable graphs and almost all perfect graphs are kernelM-solvable

1992 ◽  
Vol 8 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Mostafa Blidia ◽  
Konrad Engel
Keyword(s):  
Perfect Graphs ◽  
Almost All

On a Stein And Weiss Property of the Conjugate Function

1990 ◽  
Vol 42 (1) ◽  
pp. 109-125
Author(s):  
Nakhlé Asmar
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Almost All

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

A non-Markovian birth process with logarithmic growth

1975 ◽  
Vol 12 (04) ◽  
pp. 673-683
Author(s):  
G. R. Grimmett
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Birth Process ◽  
Image Position ◽  
Almost All ◽  
Increasing Function ◽  
Logarithmic Growth

I show that the sumof independent random variables converges in distribution when suitably normalised, so long as theXksatisfy the following two conditions:μ(n)= E |Xn|is comparable withE|Sn| for largen,andXk/μ(k) converges in distribution. Also I consider the associated birth processX(t) = max{n:Sn≦t} when eachXkis positive, and I show that there exists a continuous increasing functionv(t) such thatfor some variableYwith specified distribution, and for almost allu. The functionv, satisfiesv(t) =A(1 +o(t)) logt. The Markovian birth process with parameters λn= λn, where 0 < λ < 1, is an example of such a process.

Numbers badly approximate by fractions with prime denominator

1995 ◽  
Vol 118 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Continued Fraction ◽  
Arithmetic Progressions ◽  
The Other ◽  
Strong Result ◽  
Infinitely Many Solutions ◽  
Continued Fraction Expansion ◽  
Primes In Arithmetic Progressions ◽  
Prime Variable ◽  
Image Position ◽  
Almost All

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequalityhas infinitely many solutions if and only ifdiverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

Projection theorems for box and packing dimensions

1996 ◽  
Vol 119 (2) ◽  
pp. 287-295 ◽  
Author(s):  
K. J. Falconer ◽  
J. D. Howroyd
Keyword(s):  
Orthogonal Projection ◽  
Packing Dimension ◽  
Analytic Subset ◽  
Box Counting ◽  
Packing Dimensions ◽  
Image Position ◽  
Almost All ◽  
Projection Theorems

AbstractWe show that if E is an analytic subset of ℝn thenfor almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.

Independent dominating sets in graphs of girth five

2020 ◽  
pp. 1-16
Author(s):  
Ararat Harutyunyan ◽  
Paul Horn ◽  
Jacques Verstraete
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Regularity Conditions ◽  
Dominating Sets ◽  
Vertex Graph ◽  
First Results ◽  
Complete Bipartite Graphs ◽  
Image Position ◽  
Almost All ◽  
Independent Dominating Set

Abstract Let $\gamma(G)$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then $$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$ In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then $$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$ for some constant c independent of d. This result is sharp in the sense that as $d \rightarrow \infty$ , almost all d-regular n-vertex graphs G of girth at least five have $$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$ Furthermore, if G is a disjoint union of ${n}/{(2d)}$ complete bipartite graphs $K_{d,d}$ , then ${\gamma_\circ}(G) = \frac{n}{2}$ . We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that ${\gamma_\circ}(G) \sim {n}/{2}$ as $d \rightarrow \infty$ . Therefore both the girth and regularity conditions are required for the main result.

scholarly journals Marstrand-type Theorems for the Counting and Mass Dimensions in ℤ

2016 ◽  
Vol 25 (5) ◽  
pp. 700-743 ◽  
Author(s):  
DANIEL GLASSCOCK
Keyword(s):  
Recent Work ◽  
Orthogonal Projection ◽  
Side Length ◽  
Orthogonal Projections ◽  
Mass Dimension ◽  
Image Position ◽  
Almost All

The counting and (upper) mass dimensions of a set A ⊆ $\mathbb{R}^d$ are $$D(A) = \limsup_{\|C\| \to \infty} \frac{\log | \lfloor A \rfloor \cap C |}{\log \|C\|}, \quad \smash{\overline{D}}\vphantom{D}(A) = \limsup_{\ell \to \infty} \frac{\log | \lfloor A \rfloor \cap [-\ell,\ell)^d |}{\log (2 \ell)},$$ where ⌊A⌋ denotes the set of elements of A rounded down in each coordinate and where the limit supremum in the counting dimension is taken over cubes C ⊆ $\mathbb{R}^d$ with side length ‖C‖ → ∞. We give a characterization of the counting dimension via coverings: $$D(A) = \text{inf} \{ \alpha \geq 0 \mid {d_{H}^{\alpha}}(A) = 0 \},$$ where $${d_{H}^{\alpha}}(A) = \lim_{r \rightarrow 0} \limsup_{\|C\| \rightarrow \infty} \inf \biggl\{ \sum_i \biggl(\frac{\|C_i\|}{\|C\|} \biggr)^\alpha \ \bigg| \ 1 \leq \|C_i\| \leq r \|C\| \biggr\}$$ in which the infimum is taken over cubic coverings {Ci} of A ∩ C. Then we prove Marstrand-type theorems for both dimensions. For example, almost all images of A ⊆ $\mathbb{R}^d$ under orthogonal projections with range of dimension k have counting dimension at least min(k, D(A)); if we assume D(A) = D(A), then the mass dimension of A under the typical orthogonal projection is equal to min(k, D(A)). This work extends recent work of Y. Lima and C. G. Moreira.

scholarly journals ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT

2016 ◽  
Vol 94 (2) ◽  
pp. 224-235 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ
Keyword(s):  
Log P ◽  
Hamming Weight ◽  
Multiplicative Subgroup ◽  
Image Position ◽  
Almost All

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes $p$, there is a multiple $mp$ that can be written in binary as $$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$ with $k=6$ (corresponding to Hamming weight seven). We also prove that there are infinitely many primes $p$ with a multiplicative subgroup $A=\langle g\rangle \subset \mathbb{F}_{p}^{\ast }$, for some $g\in \{2,3,5\}$, of size $|A|\gg p/(\log p)^{3}$, where the sum–product set $A\cdot A+A\cdot A$ does not cover $\mathbb{F}_{p}$ completely.

Equidistribution of values of rational functions (mod p)

1995 ◽  
Vol 125 (5) ◽  
pp. 911-929
Author(s):  
R. W. K. Odoni ◽  
P. G. Spain
Keyword(s):  
Rational Functions ◽  
Linearly Independent ◽  
Image Position ◽  
Almost All ◽  
Mod P

Let R1(x),…, Rd(x) be rational functions in Iℚ(x), such that 1, R1(x),…, Rd(x) are linearly independent over Iℚ. For almost all primes p, their mod p reductions, are well-defined rational functions over Fp and are linearly independent over Fp We show that asymptotically the pointsare uniformly distributed in [0, l)d.

Sign in / Sign up

Export Citation Format

Share Document