The complexity of the T-coloring problem for graphs with small degree

AbstractIn the paper we consider a generalized vertex coloring model, namely T-coloring. For a given finite set T of nonnegative integers including 0, a proper vertex coloring is called a T-coloring if the distance of the colors of adjacent vertices is not an element of T. This problem is a generalization of the classic vertex coloring and appeared as a model of the frequency assignment problem. We present new results concerning the complexity of T-coloring with the smallest span on graphs with small degree Δ. We distinguish between the cases that appear to be polynomial or NP-complete. More specifically, we show that our problem is polynomial on graphs with Δ⩽2 and in the case of k-regular graphs it becomes NP-hard even for every fixed T and every k>3. Also, the case of graphs with Δ=3 is under consideration. Our results are based on the complexity properties of the homomorphism of graphs

Minimum Sum Edge Colorings of Multicycles

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength} of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems

Russia and Roman Law

The question of Russia’s European identity has traditionally been controversial. Usually, the country is either defined as belonging to Eastern Europe in a narrower sense or, contrarily, totally excluded from the concept of Europe. From the times of Czar Peter the Great (1689–1725), Russia acquired the unquestioned status of a European power; however, despite the "enlightened" reforms of Empress Catherine the Great (1762–1796), its society remained feudal, its economy backward and its government autocratic. Right up until its collapse, the Russian Empire was decidedly less urbanized and less advanced in agriculture in comparison not only with the West but also with East-Central Europe. ..

Finite-dimensionality in Tanaka theory

In this paper we extend the Tanaka finiteness theorem and inequality for the number of symmetries to arbitrary distributions (differential systems) and provide several applications.Comment: In this version I extend the last section by including a description of all rank 2 and 3 distributions with infinite-dimensional symmetry algebra. The second proof of Theorem 1 for semi-strongly regular case is omitted in this version (as excessive), but it is valid/accessible through version 1

Westen im Osten

Western European legal historiography deplores the 19th century as the age of destruction of the Romanrooted ius commune by national codes. In reality the French code civil, the Austrian ABGB and the German Pandectist jurisprudence effected rather a relative unification of private law. In the same way, French constitutionalism as well as the administrative and court system spread throughout the West. This wave of law reform also rolled over Eastern Europe, until then a patchwork of customary law. In South-Eastern Europe, previously under Byzantine influence, the Serbian Civil Code of 1844 followed the Austrian ABGB, while the Rumanian Code of 1864 followed the French code civil. All over the region the liberal Belgian constitution of 1831 was very influential. Bohemia and Poland, both of them forming the eastern periphery of the Central-European empires, simply had their law imposed upon them. Hungary and Russia, on the other hand, modernized their law mainly by means of German Pandectist jurisprudence. In this way the patchwork of eastern customary law was harmonized. The further circulation of western legal models in the interwar period completed the relative unification of continental European law. The communist rule left these common legal bases of West and East to a large extent intact