Defective 3-Paintability of Planar Graphs

A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex is uncolored and has $k$ tokens. In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.Comment: 21 pages, 11 figure

Weak degeneracy of planar graphs and locally planar graphs

Weak degeneracy is a variation of degeneracy which shares many nice properties of degeneracy. In particular, if a graph $G$ is weakly $d$-degenerate, then for any $(d + 1)$-list assignment $L$ of $G$, one can construct an $L$-coloring of $G$ by a modified greedy coloring algorithm. It is known that planar graphs of girth 5 are 3-choosable and locally planar graphs are 5-choosable. This paper strengthens these results and proves that planar graphs of girth 5 are weakly 2-degenerate and locally planar graphs are weakly 4-degenerate.Comment: 13page

Photo-enforced stratification of liquid crystal / monomer mixtures : principle, theory and analysis of a paintable LCD concept

To keep pace with customer demand for cost-effective flat panel displays, liquid crystal display (LCD) manufacturing technologies are required that enable the processing of larger substrates with increased production speeds. In cell-technology, currently used in all LCD factories, cells are formed by coupling two substrates, which are subsequently filled with liquid crystal (LC). However, this is a timeconsuming process that limits the shape- and substrate choice. New display designs require displays that are curved, optionally flexible, and non-rectangular in shape. This thesis describes the exploration of a new phase separation process that enables the production of Paintable LCDs. Unlike cell-technology, Paintable LCDs are produced on a single-substrate by the sequential coating and curing of multiple organic layers on top of each other. The electro-optical LC layer is confined between the substrate and a polymer sheet with the important feature that the latter is formed during processing. This in-situ polymer sheet formation is the result of a spatially-controlled photopolymerization-induced phase separation process. In this process, a film consisting of a mixture of LC and monomers is irradiated with UV light. Due to a photopolymerization rate gradient in the film and a concomittant selective phase separation of LC material at the bottom of the film, the film is transformed into a stratified morphology: a polymer film on top of an LC layer. This concept is referred to as photo-enforced stratification (PES). Interdigitated electrodes previously applied on the substrate switch the LC layer. The use of coating processes makes the Paintable LCD technology well suited for application in free form factor displays potentially produced via high-speed roll-toroll manufacturing processes. In PES two main physical processes are involved: photopolymerizationinduced diffusion and polymerization-induced phase separation. As a result of transversal diffusion of LC and monomers through the film, induced by a vertical gradient in the polymerization rate, the phase separation process is located at the bottom of the film. The phase separation at the bottom of the film leads to the formation of large LC domains randomly distributed over the substrate area, covered by a polymeric topcoat. Characterizations with polarization microscopy and surface profile measurements show that when the stratification step is preceded by a mask exposure step, morphologies are formed that can be described as regular arrays of neighboring LC-filled polymer capsules. Confocal Raman microscopy measurements on these LC-filled polymer capsules reveal that part of the LC stays isotropically dissolved in the polymer phase. Moreover, it was found that under the current process conditions microscopic LC droplets are formed, which are dispersed in the polymer near the polymer-LC interface, comparable to the morphology of PDLC displays (polymer-dispersed liquid crystal displays). A numerical PES model has been developed based on free radical polymerization rate equations, diffusion equations and the thermodynamics of phase separation. The PES model is a combination of two distinct components: The first component is a reaction-diffusion model that calculates the evolution of the concentration of the liquid crystal (LC), monomer and polymer as a function of depth in the film and time. The second component is a thermodynamic model that describes polymerization-induced phase separation (PIPS). In the model, the contribution of the entropic and enthalpic mixing (Flory-Huggins theory of mixing), elasticity of the polymer network (Flory-Rehner theory) and nematic ordering (Maier-Saupe theory) to the Gibbs free energy are included. The overall PES model is a one-dimensional model, which calculates the location and the time (conversion) at which the phase separation sets in. Moreover, it helps the prediction of trends in the morphologies that will be formed. In order to compare the model outcomes with the experimental results, the model input parameters have been determined, either by calculations or by experiments on the appropriate LC/monomers systems. The Flory-Huggins interaction parameters between the various components were estimated via the calculated solubility parameters of the components (via a group contribution theory). With the aid of photo-DSC the photopolymerization kinetics was investigated and the diffusion constants the LC and monomer species were measured with the aid of NMR spectroscopy as a function of the conversion. Both components of the PES model, the reaction-diffusion model and the phase separation model, were independently compared to experiments. With confocal Raman microscopy, the concentration profile of the LC compound was monitored insitu, during the UV irradiation. The measured changes in the LC concentration profile were found to be similar to the changes calculated by the reaction-diffusion model. Photo-DSC has been combined with in-situ optical microscopy to determine the phase diagram of the investigated LC/monomer/polymer system. The measurements showed that for the investigated system the elastic contribution of the polymer network could be neglected. The theoretical phase diagram, in which the phase separation lines were calculated by taking the mixing contributions and the contribution of nematic ordering of the LC phase into account, are in agreement with the experimental phase diagram. Subsequently, the stratification behavior as a function of the LC fraction in the initial reaction mixture was investigated experimentally. The earlier onset of phase separation as well as an increased formation of PDLC-morphology in the polymer layer at higher initial LC fraction both agree with trends calculated with the PES model. The PES model and the diffusion-, phase separation- and stratification experiments have led to a better understanding of the PES process in which the position of the onset of phase separation in the layer is controlled by polymerizationinduced diffusion. Besides a better understanding of the physical processes involved in the PES process this research has led to a simplified and improved stratification process in which the arrays of LC-filled polymer capsules are obtained via a single UV exposure step. For this purpose the alignment layer on the bottom substrate is first patterned with an adhesion promoter using offset printing. During the stratification process the polymer top layer locally forms covalent bonds with the adhesion promoter patterns. Besides a simpler manufacturing process, this results in mechanically stable morphologies, which enable the production of flexible, plastic LCDs with a free form factor

Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric