Some results on the palette index of graphs

Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees. Some of our results concern $(a,b)$-biregular graphs; that is, bipartite graphs where all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. We conjecture that if $G$ is $(a,b)$-biregular, then $\check{s}(G)\leq 1+\max\{a,b\}$, and we prove that this conjecture holds for several families of $(a,b)$-biregular graphs. Additionally, we characterize the graphs whose palette index equals the number of vertices

Statistical State Dynamics: a new perspective on turbulence in shear flow

Traditionally, single realizations of the turbulent state have been the object of study in shear flow turbulence. When a statistical quantity was needed it was obtained from a spatial, temporal or ensemble average of sample realizations of the turbulence. However, there are important advantages to studying the dynamics of the statistical state (the SSD) directly. In highly chaotic systems statistical quantities are often the most useful and the advantage of obtaining these statistics directly from a state variable is obvious. Moreover, quantities such as the probability density function (pdf) are often difficult to obtain accurately by sampling state trajectories even if the pdf is stationary. In the event that the pdf is time dependent, solving directly for the pdf as a state variable is the only alternative. However, perhaps the greatest advantage of the SSD approach is conceptual: adopting this perspective reveals directly the essential cooperative mechanisms among the disparate spatial and temporal scales that underly the turbulent state. While these cooperative mechanisms have distinct manifestation in the dynamics of realizations of turbulence both these cooperative mechanisms and the phenomena associated with them are not amenable to analysis directly through study of realizations as they are through the study of the associated SSD. In this review a selection of example problems in the turbulence of planetary and laboratory flows is examined using recently developed SSD analysis methods in order to illustrate the utility of this approach to the study of turbulence in shear flow.Comment: 27 pages, 18 figures. To appear in the book "Zonal jets: Phenomenology, genesis, physics", Cambridge University Press, edited by B. Galperin and P. L. Rea

The International Dimension of the Antitrust Practice in Poland, Hungary and the Czech Republic

This paper analyses how the competition authorities in the Czech republic, Poland and Hungary (CPH) have dealt with the interface between trade and competition in their actual practice. The following findings emerge (i) there has not been any significant conflict in the allocation of jurisdiction between CPH on the one hand and the EU on the other hand. This may however be due to a lack of integration between these countries. (ii) the definition of the relevant geographic market suffers from significant shortcomings in each country under review with a general bias in favour of narrow market definition. Problems are most severe in the Czech republic. (iii) Anti-trust agencies in all three countries have attempted to advocate competition in the formulation of trade policy. Developments regarding the independence of the agencies is however mixed. There are some worrying signs that the Polish agency has become less independent whereas the Hungarian agency has probably become even more independent (iv) anti-trust agencies in all three countries could indeed be pursuing objectives of industrial policy in the exercise of merger control towards foreign firms. The situation is most severe in Poland where the suspicion arises that profitable market positions have been auctioned off to foreign buyers in exchange for commitments which are unrelated to the competitive situation.antitrust; transition