scholarly journals Mr. Paint and Mrs. Correct go Fractional

10.37236/627 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Grzegorz Gutowski

We study a fractional counterpart of the on-line list colouring game "Mr. Paint and Mrs. Correct" introduced recently by Schauz. We answer positively a question of Zhu by proving that for any given graph the on-line choice ratio and the (off-line) choice ratio coincide. On the other hand it is known from the paper of Alon et al. that the choice ratio equals the fractional chromatic number. It was also shown that the limits used in the definitions of these last two notions can be realised. We show that this is not the case for the on-line choice ratio. Both our results are obtained by exploring the strong links between the on-line choice ratio, and a new on-line game with probabilistic flavour which we introduce.

2014 ◽  
Vol 4 (4) ◽  
pp. 1-16
Author(s):  
Manuel Torres ◽  
José Samos ◽  
Eladio Garví

Ontologies can be used in the construction of OLAP (On-Line Analytical Processing) systems. In such a context, ontologies are mainly used either to enrich cube dimensions or to define ontology based-dimensions. On the one hand, if dimensions are enriched using large ontologies, like WordNet, details that are beyond the scope of the dimension may be added to it. Even, dimensions may be obscured because of the massive incorporation of related attributes. On the other hand, if ontologies are used to define a dimension, it is possible that a simplified version of the ontology is needed to define the dimension, especially when the used ontology is too complex for the dimension that is being defined. These problems may be solved using one of the existing mechanisms to define ontology views. Therefore, concepts that are not needed for the domain ontology are kept out of the view. However, this view must be closed so that, no ontology component has references to components that are not included in the view. In this work, two basic approaches are proposed: enlargement and reduction closure.


10.37236/1140 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Nathan Linial ◽  
Michael Saks ◽  
David Statter

Two sets are non-crossing if they are disjoint or one contains the other. The non-crossing graph ${\rm NC}_n$ is the graph whose vertex set is the set of nonempty subsets of $[n]=\{1,\ldots,n\}$ with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover number of this graph are presented. For the chromatic number of this graph we show: $$ n(\log_e n -\Theta(1)) \le \chi({\rm NC}_n) \le n (\lceil\log_2 n\rceil-1). $$


10.37236/401 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Penny Haxell ◽  
Jacques Verstraete

Let $H$ be a hypergraph and let $L_v : v \in V(H)$ be sets; we refer to these sets as lists and their elements as colors. A list coloring of $H$ is an assignment of a color from $L_v$ to each $v \in V(H)$ in such a way that every edge of $H$ contains a pair of vertices of different colors. The hypergraph $H$ is $k$-list-colorable if it has a list coloring from any collection of lists of size $k$. The list chromatic number of $H$ is the minimum $k$ such that $H$ is $k$-list-colorable. In this paper we prove that every $d$-regular three-uniform linear hypergraph has list chromatic number at least $(\frac{\log d}{5\log \log d})^{1/2}$ provided $d$ is large enough. On the other hand there exist $d$-regular three-uniform linear hypergraphs with list chromatic number at most $\log_3 d+3$. This leaves the question open as to the existence of such hypergraphs with list chromatic number $o(\log d)$ as $d \rightarrow \infty$.


2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Yan Li ◽  
Xin Zhang

An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.


1995 ◽  
Vol 10 ◽  
pp. 585-587
Author(s):  
Keith Butler

In this paper I review some recent advances in the use of large amounts of atomic data in the modelling of atmospheres and winds of hot stars. The review is highly selective but representative of current developments. A more general overview is to be found in Kudritzki and Hummer (1990) although the field is changing so rapidly that much has happened since then. The paper breaks down into three parts: work on line formation, in which the atmospheric structure is known and held fixed, is described first, then follows a description of the inclusion of line opacities in non-LTE in the atmosphere problem itself, and finally recent developments in the theory of radiatively driven stellar winds are summarized. Here special emphasis is given to a novel distance determination method based entirely on spectroscopie quantities. I close with a brief shopping list.In a series of papers, Becker and Butler (1992,1994a, b,c) have investigated iron and nickel spectra in sub-dwarfs using the complete linearization method of Auer and Heasley (1976). The method scales linearly with the number of frequency points so they were able to use well over 10000 frequencies to adequately describe the line opacities. Several thousand lines were treated explicitly and the resultant computed spectra gave execellent fits to observed Hubble spectra in the wavelength ranges dominated by the ions concerned.The different ionization stages gave consistent results for the iron and nickel abundances but only after line-blocking from millions of spectral lines in the far UV had been included. This was done using the Kurucz (1988) line lists coupled with line grouping as suggested by Anderson (1989) and described briefly in the next section.The line-blanketed atmospheres of Kurucz (1991) are the best available up to about 30000K, where non-LTE effects start to become important. Non-LTE line-blanketed atmospheres have become feasible because the computational requirements of the accelerated lambda iteration (ALI) method (Werner and Husfeld, 1985) also scale linearly with the number of frequency points. On the other hand, Anderson (1989) suggested grouping energetically adjacent atomic levels together to form pseudo-levels on the basis that although they might, as a group, be in non-LTE, they should be in LTE with respect to one another due to the large number of collisions between them. This greatly reduces the number of levels to be considered but instead gives rise to highly complicated pseudo line-profiles. Grigsby et al (1992), who did not use ALI, constructed the first grid of line-blanketed non-LTE models by using a variation on the Opacity Distribution Function concept to group line opacities into blocks thereby reducing the number of frequency points required. Dreizler and Werner (1993) on the other hand were able to sample the opacity as they used ALI in their models.


10.37236/3228 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Naoki Matsumoto

A graph $G$ is uniquely $k$-colorable if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $e\in E(G)$. In this paper, we prove that if $G$ is an edge-critical uniquely $3$-colorable planar graph, then $|E(G)|\leq \frac{8}{3}|V(G)|-\frac{17}{3}$. On the other hand, there exists an infinite family of edge-critical uniquely 3-colorable planar graphs with $n$ vertices and $\frac{9}{4}n-6$ edges. Our result gives a first non-trivial upper bound for $|E(G)|$.


10.37236/999 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Ashley Montanaro ◽  
Michael W. Newman ◽  
Simone Severini ◽  
Andreas Winter

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph.After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is $2$, nor if it is $3$ in a restricted quantum model; on the other hand, we exhibit a graph on $18$ vertices and $44$ edges with chromatic number $5$ and quantum chromatic number $4$.


10.37236/5003 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Alan Frieze ◽  
Dieter Mitsche ◽  
Xavier Pérez-Giménez ◽  
Paweł Prałat

In this paper, the on-line list colouring of binomial random graphs $\mathcal{G}(n,p)$ is studied. We show that the on-line choice number of $\mathcal{G}(n,p)$ is asymptotically almost surely asymptotic to the chromatic number of $\mathcal{G}(n,p)$, provided that the average degree $d=p(n-1)$ tends to infinity faster than $(\log \log n)^{1/3} (\log n)^2 n^{2/3}$. For sparser graphs, we are slightly less successful; we show that if $d \ge (\log n)^{2+\epsilon}$ for some $\epsilon>0$, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of $C$, where $C \in [2,4]$, depending on the range of $d$. Also, for $d=O(1)$, the on-line choice number is by at most a multiplicative constant factor larger than the chromatic number.


Author(s):  
Barbara Lena Gierszewska

Digita l cinema history should support the conduct of free research, encourage authors to share the results of their findings and source discoveries. The problem is that Polish digital history of cinema on one hand is very poor (still a small part of sources and theoretical, historical and critical scientifi cachievements is available on-line), on the other hand, what has already appeared, is not representative because it omits largely the work of film studying authorities. For example, on the Internet you can find the views of researchers of the younger generation about the cultural history of the cinema in Polish context, including criticism of thesis described by scientifictycoons, but these are the texts that can be read only in paper publications. It begins to be, by the scientific standards, strange, because in the history of digital cinema leading roles are played by the young researchers, but without a clash of views with the authorities. The fact that on the net there are no statements concerning the scientifi cresearch given by the greatest experts of Polish cinema, who do not want to turn into a discussion with the young ones, causes the situation in which data to the history of Polish cinema digitally available are (so far) not always satisfying either as the aspect of cognition or the source.  


Author(s):  
Kenhaiya L. Bardia ◽  
Donald G. LaBounty ◽  
Timothy Breig

Paragraph UG-28 of the ASME Section VIII, Division 1 provides rules for reinforcement of the cone-to-cylinder junction based on line of support criteria. Code Case 2286, on the other hand, provides rules based on actual stresses versus allowable stresses. In this paper, the authors compare the results of the calculations for the reinforcement of the cone-to-cylinder junction for each method. It was found that, in some cases, the reinforcement method of UG-28 will produce non-conservative results.


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