Monochromatic Components Research Articles

Islands in Graphs on Surfaces

An island in a graph is a set $X$ of vertices, such that each element of $X$ has few neighbors outside $X$. In this paper, we prove several bounds on the size of islands in large graphs embeddable on fixed surfaces. As direct consequences of our results, we obtain that: (1) Every graph of genus $g$ can be colored from lists of size 5, in such a way that each monochromatic component has size $O(g)$. Moreover all but $O(g)$ vertices lie in monochromatic components of size at most 3. (2) Every triangle-free graph of genus $g$ can be colored from lists of size 3, in such a way that each monochromatic component has size $O(g)$. Moreover all but $O(g)$ vertices lie in monochromatic components of size at most 10. (3) Every graph of girth at least 6 and genus $g$ can be colored from lists of size 2, in such a way that each monochromatic component has size $O(g)$. Moreover all but $O(g)$ vertices lie in monochromatic components of size at most 16. While (2) is optimal up to the size of the components, we conjecture that the size of the lists can be decreased to 4 in (1), and the girth can be decreased to 5 in (3). We also study the complexity of minimizing the size of monochromatic components in 2-colorings of planar graphs.

Tractability and hardness of flood-filling games on trees

This work presents new results on flood-filling games, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. A flooding move consists of changing the color of the monochromatic component containing a vertex p (the pivot of the move). These games are originally played on grids; however, when played on trees, we have interesting applications and significant effects on problem complexity. In this paper, a complete mapping of the complexity of flood-filling games on trees is made, charting the consequences of single and aggregate parameterizations by: number of colors, number of moves, maximum distance from the pivot, number of occurrences of a color, number of leaves, and difference between number of moves and number of colors.We show that Flood-It on trees and Restricted Shortest Common Supersequence (RSCS) are analogous problems, in the sense that they can be translated from one to another, preserving complexity issues; this implies interesting FPT and W[1]-hard cases to Flood-It. Restricting attention to phylogenetic colored trees (where each color occurs at most once in any root-leaf path, in order to model phylogenetic sequences), we also show some impressive NP-hard and FPT results for both games. In addition, we prove that Flood-It and Free-Flood-It remain NP-hard when played on 3-colored trees, which closes an open question posed by Fleischer and Woeginger. Finally, we present a general framework for reducibility from Flood-It to Free-Flood-It; some NP-hard cases for Free-Flood-It can be derived using this approach.

On the temperature dependence of silicon quantum dot photoluminescence

A model of radiative and nonradiative transitions in silicon quantum dots is presented, which describes the photoluminescence temperature dependence of the ion-synthesized ensembles of Si nanocrystals in SiO2. A four-level system of transitions is considered that takes into account thermally activated processses and exchange splitting of the ground energy state of exciton in a silicon nanocrystal into triplet and singlet levels, the transfers from which to the ground state are responsible for the luminescence. The temperature dependence of monochromatic photoluminescence components was obtained based on the stationary solution of the system of kinetic equations for the population of levels that satisfactory describes the experimental data. The exciton energy splitting values depending on the energy of emitted photons were found and compared with the data in the literature.

A Note on Covering Edge Colored Hypergraphs by Monochromatic Components

For $r\geq 2$, $\alpha \geq r-1$ and $k\geq 1$, let $c(r,\alpha ,k)$ be the smallest integer $c$ such that the vertex set of any non-trivial $r$-uniform $k$-edge-colored hypergraph ${\cal H}$ with $\alpha ({\cal H})=\alpha$ can be covered by $c$ monochromatic connected components. Here $\alpha({\cal{H}})$ is the maximum cardinality of a subset $A$ of vertices in $\cal{H}$ such that $A$ does not contain any edges. An old conjecture of Ryser is equivalent to $c(2,\alpha,k)=\alpha (r-1)$ and a recent result of Z. Király states that $c(r,r-1,k)=\lceil \frac{k}{r}\rceil$ for any $r\ge 3$.Here we make the first step to treat non-complete hypergraphs, showing that $c(r,r,r)=2$ for $r\ge 2$ and $c(r,r,r+1)=3$ for $r\ge 3$.

Colouring Planar Graphs With Three Colours and No Large Monochromatic Components

We prove the existence of a function$f :\mathbb{N} \to \mathbb{N}$such that the vertices of every planar graph with maximum degree Δ can be 3-coloured in such a way that each monochromatic component has at mostf(Δ) vertices. This is best possible (the number of colours cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by Kleinberg, Motwani, Raghavan and Venkatasubramanian in 1997. Our result extends to graphs of bounded genus.

Gallai-Colorings of Triples and 2-Factors of

A coloring of the edges of the -uniform complete hypergraph is a -coloring if there is no rainbow simplex; that is, every set of vertices contains two edges of the same color. The notion extends -colorings which are often called Gallai-colorings and originates from a seminal paper of Gallai. One well-known property of -colorings is that at least one color class has a spanning tree. J. Lehel and the senior author observed that this property does not hold for -colorings and proposed to study , the size of the largest monochromatic component which can be found in every -coloring of , the complete -uniform hypergraph. The previous remark says that and in this note, we address the case . We prove that and this determines for . We also prove that by excluding certain 2-factors from the middle layer of the Boolean lattice on seven elements.

Monochromatic components in edge-colored complete uniform hypergraphs

Let Knr denote the complete r-uniform hypergraph on vertex set V=[n]. An f-coloring is a coloring of the edges with colors {1,2,…,f}; it defines monochromatic r-uniform hypergraphs Hi=(V,Ei) for i=1,…,f, where Ei contains the r-tuples colored by i. The connected components of hypergraphs Hi are called monochromatic components. For n>rk let f(n,r,k) denote the maximum number of colors, such that in any f-coloring of Knr, there exist k monochromatic components covering V. Moreover let f(r,k)=minn>rkf(n,r,k). A reformulation (see Gyárfás (1977) [5]) of an important special case of Ryser’s conjecture states that f(2,k)=k+1 for all k. This conjecture is proved to be true only for k≤4, so the value of f(2,5) is not known. On the contrary, in this paper we show that for r>2 we can determine f(r,k) exactly, and its value is rk.

Application of empirical mode decomposition to a Jeffcott rotor with a breathing crack

Fatigue damage, appearing due to developing cracks, is considered to be one of the main faults in rotating machinery. The damage in rotating components can be catastrophic, posing a potential hazard and leading to significant economic loss. In the related literature, transverse breathing cracks are considered as a primary mode of damage. Accordingly, a Jeffcott rotor with a transverse breathing crack has been examined here wherein a method for identifying the early crack propagation is proposed. The breathing functions developed in a recent publication to approximate the actual breathing mechanism of the cracked shaft are employed along with the method of empirical mode decomposition (EMD) to identify the crack vibration signature. EMD combined with the wave transform spectrum is used to decompose the measured vibration time series of the cracked rotor system into nearly monochromatic components. It is shown that the variations of the averaged amplitudes of the super-harmonic components in the neighborhood of 1/2 and 1/3 of the first critical rotational speed provide clear and robust vibration signatures indicating the early presence of the breathing crack. This signature is utilized here for early crack detection in the rotor system under consideration.

Model of photoluminescence from ion-synthesized silicon nanocrystal arrays embedded in a silicon dioxide matrix

A four-level model of photoluminescence from Si nanocrystal arrays embedded in a SiO2 matrix is suggested. The model allows for thermally activated transitions between singlet and triplet levels in the exchange-split energy state of an exciton in an excited silicon nanocrystal. An expression is derived for the temperature dependence of the intensity of photoluminescence monochromatic components. A correlation is found between the amount of splitting and the emitted photon energy by comparing model data with our experimental data for ion-synthesized Si nanocrystals in a SiO2 matrix. The model explains the finiteness of the photoluminescence intensity at temperatures close to 0 K and the nonmonotonicity of the temperature run of the intensity.

Determination of the spatial dependence of the complex components of the tensor $$\hat \chi ^{(2)}$$ (z, ω 1 + ω 2; ω 1, ω 2) of a medium with one-dimensional inhomogeneity using a biharmonic wave

It is proven that the spatial profiles of different components of the complex quadratic susceptibility tensor \(\hat \chi ^{(2)}\) (z,ω1+ω2; ω1, ω2), which is responsible for the generation of sum-frequency wave in a plate with one-dimensional inhomogeneity, can be reconstructed unambiguously. A reconstruction technique is proposed. To implement it, one has to direct a plane biharmonic wave with monochromatic components at frequencies ω1 and ω2 onto a plate and measure (in some range of the angles of incidence) the complex amplitude of the sum-frequency wave reflected from the plate. Changing the plane of incidence of the initial wave and (or) the polarization of its monochromatic components, one can determine the coordinate dependences for more than half of the components of \(\hat \chi ^{(2)}\) (z,ω1+ω2; ω1, ω2). This reconstruction can be performed if the symmetry of the plate medium provides a diagonal form for its linear permittivity tensor. The technique proposed implies measurement of the intensities of the sum-frequency waves generated under special conditions using an auxiliary reference plate; this approach allows one to do without complex phase measurements.

Mapping of the second-order nonlinear susceptibility of inhomogeneous absorbing media by Maker fringes analysis of optical difference mixing

This paper proves the possibility of and offers a technique for a unique reconstitution of the spatial profiles of various components of the complex second-order susceptibility tensor χ^(2)(z,ω1-ω2;ω1,-ω2) responsible for difference frequency generation in one-dimensionally inhomogeneous plates. To be able to achieve such reconstitution it is necessary to measure the complex amplitude of the reflected difference frequency wave in a certain range of incidence angles of the plane biharmonic wave with frequencies ω1 and ω2 in its spectrum. By altering the incidence plane of the initial biharmonic wave and/or the polarization of its monochromatic components it is possible to uniquely determine the coordinate dependences of more than a half of all second-order susceptibility tensor components. Such reconstitution is successful when the symmetric nature of the medium ensures the diagonal character of the linear dielectric permittivity tensor of the plate. Besides, the dielectric properties of the plate can change arbitrarily along the z axis. The suggested approach includes measuring the intensity of difference frequency waves generated under specific conditions with the use of an auxiliary reference plate to be able to avoid complex phase measurements.

Monochromatic components in edge-colored complete uniform hypergraphs

Abstract Let K n r denote the complete r-uniform hypergraph on vertex set V = [ n ] . An f-coloring is a coloring of the edges with colors { 1 , 2 , … , f } , it defines monochromatic r-uniform hypergraphs H i = ( V , E i ) for i = 1 , … , f , where E i contains the r-tuples colored by i. The connected components of hypergraphs H i are called monochromatic components. For n > r k let f ( n , r , k ) denote the maximum number of colors, such that in any f-coloring of K n r , there exist k monochromatic components covering V. Moreover let f ( r , k ) = min n > r k f ( n , r , k ) . A reformulation (see [A. Gyarfas Partition coverings and blocking sets in hypergraphs, Commun. Comput. Autom. Inst. Hungar. Acad. Sci. 71 (1977)]) of an important special case of Ryserʼs conjecture states that f ( 2 , k ) = k + 1 for all k. This conjecture is proved to be true only for k ⩽ 4 , so the value of f ( 2 , 5 ) is not known. On the contrary, in this paper we show that for r > 2 we can determine f ( r , k ) exactly, and its value is rk.

Large components in r‐edge‐colorings of Kn have diameter at most five

AbstractReflecting on problems posed by Gyárfás [Ramsey Theory Yesterday, Today and Tomorrow, Birkhäuser, Basel, 2010, pp. 77–96] and Mubayi [Electron J Combin 9 (2002), #R42], we show in this note that every r‐edge‐coloring of Kn contains a monochromatic component of diameter at most five on at least n/(r−1) vertices. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 337–340, 2012

Coloring the Edges of a Random Graph without a Monochromatic Giant Component

We study the following two problems: i) Given a random graph $G_{n, m}$ (a graph drawn uniformly at random from all graphs on $n$ vertices with exactly $m$ edges), can we color its edges with $r$ colors such that no color class contains a component of size $\Theta(n)$? ii) Given a random graph $G_{n,m}$ with a random partition of its edge set into sets of size $r$, can we color its edges with $r$ colors subject to the restriction that every color is used for exactly one edge in every set of the partition such that no color class contains a component of size $\Theta(n)$? We prove that for any fixed $r\geq 2$, in both settings the (sharp) threshold for the existence of such a coloring coincides with the known threshold for $r$-orientability of $G_{n,m}$, which is at $m=rc_r^*n$ for some analytically computable constant $c_r^*$. The fact that the two problems have the same threshold is in contrast with known results for the two corresponding Achlioptas-type problems.

On the maximal temporal amplitude of down stream running nonlinear water waves

This paper concerns with the down-stream propagation of waves over initially still water. Such a study is relevant to generate waves of large amplitude in wave tanks of a hydrodynamic laboratory. Input in the form of a time signal is provided at the wave-maker located at one side of the wave tank; the resulting wave then propagates over initially still water towards the beach at the other side of the tank. Experiments show that nonlinear effects will deform the wave and may lead to large waves with wave heights larger than twice the original input; the deformations may show itself as peaking and splitting. It is of direct scientific interest to understand and quantify the nonlinear distortion; it is also of much practical interest to know at which location in the wave tank, the extreme position, the waves will achieve their maximum amplitude and to know the amplitude amplification factor. To investigate this, a previously introduced concept called Maximal Temporal Amplitude (MTA) is used: at each location the maximum over time of the wave elevation. An explicit expression of the MTA cannot be found in general from the governing equations and generating signal. In this paper we will use a Korteweg - de Vries (KdV) model and third order approximation theory to calculate the approximate extreme positions for two classes of waves. The classes are the wave-groups that originate from initially bi-chromatics and Benjamin-Feir (BF) type of waves, described by superposition of two or three monochromatic waves. We show that for initially bi-chromatics signals, the extreme position does not depend on the phases of the mono-chromatic components. For BF signals, however, the phases of the mono-chromatic components influence the extreme position essentially. The theoretical results are verified for the case of bi-chromatics with numerical as well as experimental results; for BF signals we use an analytical solution called the Soliton on Finite Background (SFB) for comparison.

Spectrally partially coherent propagation-invariant fields

Polychromatic propagation-invariant fields (localized waves) with variable spectral coherence properties are considered. A bridge is provided between coherent propagation-invariant pulses, including Bessel pulses, X-waves and focus wave modes, and stationary polychromatic propagation-invariant fields using a difference-frequency spectral-correlation model. In the space-time domain, this assumption leads to a representation of a partially coherent field as a continuum of spectrally fully coherent, mutually independent pulses originating at different instants of time. The influence of partial spectral coherence on temporal coherence and pulse shape as well as intensity distribution and transverse spatial coherence is illustrated by considering fields with propagation-invariant Bessel-mode monochromatic components.

Boundary element modeling of multidirectional random waves in a harbor with a rectangular navigation channel

A numerical model is presented for the prediction of the wave field due to the diffraction of directional random waves in a harbor with a long navigation channel dredged in the exterior region of the harbor entrance. In this study, partially reflected boundaries are considered in the harbor region and different constant water depths are set at the channel, the harbor and the exterior region of the study domain. The method for simulating the directional random waves is based on the superposition of diffraction solutions for the monochromatic wave components obtained by a two-dimensional boundary integral approach. The incident wave conditions are specified using discretized forms of the Mitsuyasu's frequency spectrum and directional function. The present numerical model is validated through comparisons with published experimental data, where good agreement between the simulated results and experimental data are obtained. Other example calculations to examine the effect of long rectangular channels (or pits) on the wave field in the harbors are also presented. Based on the comparisons in results, it is concluded that the present numerical model is an adequate and efficient tool to predict the wave field inside a harbor and around a pit like navigation channel for many practical applications.

Coloring the edges of a random graph without a monochromatic giant component

Abstract Our goal is to color the edges of a random graph G n , m (a graph drawn uniformly at random from all graphs on n vertices with exactly m edges) with a fixed number r of colors such that no color class induces a component of size Ω ( n ) – a so called ‘giant component’. We prove that for every r ⩾ 2 there exists an analytically computable constant c r ∗ for which the following holds: For any c c r ∗ , with probability 1 − o ( 1 ) there exists an r-edge-coloring of G n , rcn in which every monochromatic component has sublinearly many vertices. On the other hand, for any c > c r ∗ , with probability 1 − o ( 1 ) every r-edge-coloring of G n , rcn contains a monochromatic component on linearly many vertices. In other words, we prove that the property in question has a sharp threshold at m = r c r ∗ n .

Phase and Polarization State of High-Frequency Relic Gravitational Waves

The displaying condition of strength, phase and polarization states of high-frequency relic gravitational waves (HFRGWs) in electromagnetic (EM) detecting systems is studied. It is shown that the displaying condition depends not only on the sensitivity of EM detecting systems and the amplitudes of HFRGWs, but also on the phase, the polarization states of HFRGWs and their matching to the EM detecting systems. In order to display simultaneously the strength, phase and polarization states of the resonant “monochromatic component" of HFRGWs, an important necessary condition is the utilization of two or more different EM detectors.

Video-chromatic measurements of temperature fields during combustion of low-temperature solid propellants

A method for measuring temperature fields in luminous zones of burning nonmetal condensed substances (sooty flames) was implemented in the form of an automated information-measuring system which provides computer processing of digital video images with separation of the video signal into monochromatic components. The system was used to study the combustion of a low-temperature solid-propellant composition with a coolant. The statistical parameters of the temperature fields were estimated by stochastic modeling of video-recorded information, providing a better insight into the hotspot-pulsating combustion mechanism of such compositions.