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## A tractable NP-completeness proof for the two-coloring without monochromatic cycles of fixed length

For any integer k>2, we consider the following decision problem. Given a simple graph, does there exist a partition of its vertices into two disjoint sets such that every simple k-cycle of G contains vertices in both of these sets? This problem is NP-hard because it admits a polynomial reduction from NAE 3-SAT. We construct a reduction that is polynomial both in the length of the instance and in k, which answers a recent question of Karpinski.

The book contains the necessary information from the algorithm theory, graph theory, combinatorics. It is considered partially recursive functions, Turing machines, some versions of the algorithms (associative calculus, the system of substitutions, grammars, Post's productions, Marcov's normal algorithms, operator algorithms). The main types of graphs are described (multigraphs, pseudographs, Eulerian graphs, Hamiltonian graphs, trees, bipartite graphs, matchings, Petri nets, planar graphs, transport nets). Some algorithms often used in practice on graphs are given. It is considered classical combinatorial configurations and their generating functions, recurrent sequences. It is put in a basis of the book long-term experience of teaching by authors the discipline «Discrete mathematics» at the business informatics faculty, at the computer science faculty* *of National Research University Higher School of Economics, and at the automatics and computer technique faculty of National research university Moscow power engineering institute. The book is intended for the students of a bachelor degree, trained at the computer science faculties in the directions 09.03.01 Informatics and computational technique, 09.03.02 Informational systems and technologies, 09.03.03 Applied informatics, 09.03.04 Software Engineering, and also for IT experts and developers of software products.

Graph coloring problem is one of the classical combinatorial optimization problems. This problem consists in finding the minimal number of colors in which it is possible to color vertices of a graph so that any two adjacent vertices are colored in different colors. The graph coloring problem has a wide variety of applications including timetabling problems, processor register allocation problems, frequency assignment problems, data clustering problems, traffic signal phasing problems, maximum clique problem, maximum independent set problem, minimum vertex cover problem and others. In this paper a new efficient heuristic algorithm for the graph coloring problem is presented. The suggested algorithm builds the same coloring of a graph as does the widely used greedy sequential algorithm in which at every step the current vertex is colored into minimal feasible color. Computational experiments show that the presented algorithm performs graph coloring much faster in comparison with the standard greedy algorithm. The speedup reaches 5,6 times for DIMACS graphs.

In this chapter, we present our enhancements of one of the most efficient exact algorithms for the maximum clique problem—MCS algorithm by Tomita, Sutani, Higashi, Takahashi and Wakatsuki (in Proceedings of WALCOM’10, 2010, pp. 191–203). Our enhancements include: applying ILS heuristic by Andrade, Resende and Werneck (in Heuristics 18:525–547, 2012) to find a high-quality initial solution, fast detection of clique vertices in a set of candidates, better initial coloring, and avoiding dynamic memory allocation. A good initial solution considerably reduces the search tree size due to early pruning of branches related to small cliques. Fast detecting of clique vertices is based on coloring. Whenever a set of candidates contains a vertex adjacent to all candidates, we detect it immediately by its color and add it to the current clique avoiding unnecessary branching. Though dynamic memory allocation allows to minimize memory consumption of the program, it increases the total running time. Our computational experiments show that for dense graphs with a moderate number of vertices (like the majority of DIMACS graphs) it is more efficient to store vertices of a set of candidates and their colors on stack rather than in dynamic memory on all levels of recursion. Our algorithm solves p_hat1000-3 benchmark instance which cannot be solved by the original MCS algorithm. We got speedups of 7, 3000, and 13000 times for gen400_p0.9_55, gen400_p0.9_65, and gen400_p0.9_75 instances, correspondingly.

In the paper, we address mission critical systems, such as automobile, avionic, mobile robotic, telecommunication, etc. Such systems must meet hard real-time constraints in order to avoid catastrophic consequences. To meet the real-time constraints, strict periodicity is used (i.e. for any periodic task, time between release points is constant). Sensors, actuators and feedback control functions are typical examples of strict periodic tasks. We study a monoprocessor preemptive scheduling problem for arbitrary number of strict periodic tasks. In this context, we focus on the following problem: how to find non-conflict set of task release points (i.e. sequences of instance release points for different tasks must not intersect). First, as a preliminaries, we introduce some fundamental definitions and prove several elementary schedulability conditions. Next, we investigate the correlation between the scheduling problem and a graph coloring problem for graphs of some special kinds. The graphs under consideration are built on the basis of the tasks' period values. We introduce a notion of divisibility graph for tasks' periods, and study compatibility of graphs' coloring with respect to the schedulability problem. At last, we prove a theorem that provides necessary and sufficient graph coloring conditions for schedulability of given strict periodic tasks. This theorem allows either to find non-conflict set of task release points directly, or to determine quickly that scheduling is impossible.

This paper is devoted to the study of the graph sequence Gn = (Vn, En), where Vn is the set of all vectors v ∈ R n with coordinates in {−1, 0, 1} such that |v| = √ 3 and En consists of all pairs of vertices with scalar product 1. We find the exact value of the independence number of Gn. As a corollary we get new lower bounds on χ(R n ) and χ(Qn ) for small values of n.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.