Asymmetric coloring games on incomparability graphs

Consider the following game on a graph $G$: Alice and Bob take turns coloring the vertices of $G$ properly from a fixed set of colors; Alice wins when the entire graph has been colored, while Bob wins when some uncolored vertices have been left. The game chromatic number of $G$ is the minimum number of colors that allows Alice to win the game. The game Grundy number of $G$ is defined similarly except that the players color the vertices according to the first-fit rule and they only decide on the order in which it is applied. The $(a,b)$-game chromatic and Grundy numbers are defined likewise except that Alice colors $a$ vertices and Bob colors $b$ vertices in each round. We study the behavior of these parameters for incomparability graphs of posets with bounded width. We conjecture a complete characterization of the pairs $(a,b)$ for which the $(a,b)$-game chromatic and Grundy numbers are bounded in terms of the width of the poset; we prove that it gives a necessary condition and provide some evidence for its sufficiency. We also show that the game chromatic number is not bounded in terms of the Grundy number, which answers a question of Havet and Zhu

On-line partitioning of width w posets into w^O(log log w) chains

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of a presented poset.Comment: 16 pages, 10 figure

Coloring triangle-free rectangle overlap graphs with O(log⁡log⁡n)O(\log\log n) colors

Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $\Theta(\log\log n)$. Essentially the same construction produces $\Theta(\log\log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in $\mathbb{R}^2$ closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles, which can be alternatively described as overlap graphs of axis-parallel rectangles. That is, we prove that triangle-free rectangle overlap graphs have chromatic number $O(\log\log n)$, improving on the previous bound of $O(\log n)$. To this end, we exploit a relationship between off-line coloring of rectangle overlap graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem. Then, these subgraphs are colored with $O(\log\log n)$ colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).Comment: Minor revisio

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

An easy subexponential bound for online chain partitioning

Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width $w$ into $w^{14\lg w}$ chains. We improve this to $w^{6.5 \lg w + 7}$ with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach.Comment: 23 pages, 11 figure

Role of gut microbiota in pathogenesis of selected chronic diseases

The human digestive system is colonized by a huge number of microorganisms, that are referred to collectively as the gut microbiota. The composition of intestinal microorganisms are shaped from an early life and undergoes constant changes depending on the influence of external factors, such as: type of delivery, feeding the young child, diet in subsequent years of life, pharmaceuticals use, stress, lifestyle or infections and previous inflammation within the digestive tract. Despite transient changes in microbiota composition, the intestinal ecosystem is constantly striving to maintain homeostasis, both qualitative and quantitative, which is fundamental to human health and human development. Microbes present in the intestines are responsible for sealing the intestinal barrier, mucin production, stimulation of the angiogenesis process, supporting digestive processes by fermentation and decomposition of undigested food residues, vitamin production or protection from pathogenic microorganisms. As shown by numerous studies carried out in recent years, intestinal dysbiosis plays a fundamental role in the development of many chronic diseases such as inflammatory bowel disease, diabetes, obesity, celiac disease, connective tissue diseases and others. Insightful understanding of the interactions between microorganisms and the host organisms can provide new information about pathogenesis of diseases as well as new ways to prevent and treat intestinal or systemic disorders. The aim of this work is to review the latest reports on the role of the gastrointestinal microbiome in selected chronic diseases

Krytyka kapitalizmu i liberalnej demokracji z perspektywy współczesnego buddyzmu. Wybrane przykłady tajskiej myśli społecznej

The article describes chosen concepts of Thai Buddhist thought, which is critical to liberal democracy and capitalist order. The examples of the thought are: Buddhadasa Bhikkhu's conception of "dhammic socialism"; the idea of "middle - way economics" forcing by Prayudh Payutto and Sulak Sivaraksa's thesis on balanced economics and moral governments. All three authors mentioned above refer to the doctrinal basis of the modern Theravada Buddhism mainstream in Thailand as well as they are recognized as the representatives of "Socially engaged Bhuddism".Artykuł omawia wybrane koncepcje tajskiej myśli buddyjskiej, krytycznej wobec liberalnej demokracji oraz kapitalistycznego porządku ekonomicznego. Przykładami tej myśli są: koncepcja "dhamicznego socjalizmu" Buddhadasy bhikkhu, założenia "ekonomii środka" Prayudha Payutto oraz tezy na temat zrównoważonej gospodarki i moralnych rządów Sulaka Sivaraksy. Wszyscy wymienieni autorzy bazują na doktrynalnych podstawach głównego nurtu współczesnego buddyzmu theravada w Tajlandii oraz są uznawani za reprezentantów "buddyzmu społecznie zaangażowanego"