Tree-complete graph ramsey numbers

The ramsey number of any tree of order m and the complete graph of order n is 1 + (m ? 1)(n ? 1).     

Distances in orientations of graphs

We prove that there is a function h(k) such that every undirected graph G admits an orientation H with the following property: If an edge uv belongs to a cycle of length k in G, then uv or vu belongs to a directed cycle of length at most h(k) in H. Next, we show that every undirected bridgeless graph of radius r admits an orientation of radius at most r² + r, and this bound is best possible. We consider the same problem with radius replaced by diameter. Finilly, we show that the problem of deciding whether an undirected graph admits an orientation of diameter (resp. radius) 2 belongs to a class of problems called NP-hard.     

Tough graphs and hamiltonian circuits

The toughness of a graph G is defined as the largest real number t such that deletion of any s points from G results in a graph which is either connected or else has at most s/t components. Clearly, every hamiltonian graph is 1-tough. Conversely, we conjecture that for some t₀, every t₀-tough graph is hamiltonian. Since a square of a k-connected graph is always k-tough, a proof of this conjecture with t₀ = 2 would imply Fleischner's theorem (the square of a block is hamiltonian). We construct an infinite family of ($\text{3}\text{2}$)-tough nonhamiltonian graphs.     

Chromatic automorphisms of graphs

The vertex-coloring group and the full automorphism group of an n-colorable graph are independent if and only if n is an integer ≥ 3.     

Effects of molecular water models on Molecular Dynamic simulation of nanoflows

We study an influence of the effect of water molecular models and thermostats on approaching equilibrium in Molecular Dynamics simulation of the nanoflows of water. Out of large number of existing models of water we present the results for four selected ones, i.e.: TIP4P, PPC, TIP4P-2005, TIP5P, and for two thermostats - Gaussian and Nosé-Hoover. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The space of -limit sets of a continuous map of the interval

We first give a geometric characterization of -limit sets. We then use this characterization to prove that the family of -limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems.

     

Worst scenario method in homogenization. Linear case

The paper deals with homogenization of a linear elliptic boundary problem with a specific class of uncertain coefficients describing composite materials with periodic structure. Instead of stochastic approach to the problem, we use the worst scenario method due to Hlaváček (method of reliable solution). A few criterion functionals are introduced. We focus on the range of the homogenized coefficients from knowledge of the ranges of individual components in the composite, on the values of generalized gradient in the places where these components change and on the average of homogenized solution in some critical subdomain. This research was supported by grant No. 201/03/0570 of the Grant Agency of the Czech Republic.