Ivan Havel

SCALE DIMENSIONS IN NATURE

The aim of this study is to introduce and explain two dimensions of reality which, insofar as they structure thought, are typically excluded from our thought contents. The first is related to the order of spatial scales and the second is related to the order of temporal scales. We introduce two additional coordinate axes, one for the spatial scales and one for the temporal scales, to represent these orders. Various geometrical, physical, and systemic aspects of these new dimensions are treated, as well as the related concept of (a hierarchy of) functional or descriptional levels and the role of interaction between them. Using the scale perspective may contribute to developing a model of consciousness that is at once specific and allows for the infinite complexity and subjective contents which seem to undermine purely mechanistic models of consciousness.

The diameter of the cube-connected cycles

Cube-connected cycles, or CCC, are graphs with properties which make them possible candidates for switching patterns of multiprocessor computers. In this paper, the diameter of CCC is calculated. In fact, the same calculation works for somewhat more general graphs than just CCC.

Spanning caterpillars of a hypercube

Spanning trees of the hypercube Qn have been recently studied by several authors. In this paper, we construct spanning trees of Qn which are caterpillars and establish quantitative bounds for a caterpillar to span Qn. As a corollary, we disprove a conjecture of Harary and Lewinter on the length of the spine of a caterpillar spanning Qn.

Spherical and Clockwise Spherical Graphs

The main subject of our study are spherical (weakly spherical) graphs, i.e. connected graphs fulfilling the condition that in each interval to each vertex there is exactly one (at least one, respectively) antipodal vertex. Our analysis concerns properties of these graphs especially in connection with convexity and also with hypercube graphs. We deal e.g. with the problem under what conditions all intervals of a spherical graph induce hypercubes and find a new characterization of hypercubes: G is a hypercube if and only if G is spherical and bipartite.

Routing permutations and 2---1 routing requests in the hypercube

Let Hn be the directed symmetric n-dimensional hypercube. Using the computer, we show that for anypermutation of the vertices of H4, there exists a system of pairwise arc-disjoint directed paths from each vertex to its target in the permutation. This veri3es Szymanski’s conjecture (Proceedings of the International Conference on Parallel Processing, 1989, pp. I-103–I-110) for n = 4. We also consider the so-called 2–1 routing requests in Hn, where anyvertex can be used twice as a source but onlyonce as a target; we construct for any n?3 a 2–1 request that cannot be routed in Hn byarc-disjoint paths: in other words, for n?3, Hn is not (2–1)-rearrangeable. ? 2001 Elsevier Science B.V. All rights reserved.

On the Way to Intelligence Singularity

Since the fifties of the last century there have been debates about the so called “technological singularity”, motivated by the predicted and later actual exponential growth of the speed and power of computers. Recently the interest of futurologists and philosophers shifts to the so called ‘intelligence singularity’ which some of them predict to happen soon after human intelligence is surpassed by artificial intelligence. This study critically analyzes certain assumptions behind the concept of intelligence singularity, in particular the idea of explosive growth of intelligence of machines with the ability of designing machines more intelligent than themselves.

On paths and cycles dominating hypercubes

The aim of the present paper is to study the properties of the hypercube related to the concept of domination. We derive upper and lower bounds and prove an interpolation theorem for related invariants.

On 2-periodic graphs of a certain graph operator

We deal with the graph operator Pow2 defined to be the complement of the square of a graph: Pow2(G) = Pow2(G). Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class G of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph Km,n can be decomposed in two edge-disjoint factors from G. We further show that all the incidence graphs of Desarguesian finite projective geometries belong to G and find infinitely many graphs also belonging to G among generalized hypercubes.

Euler Cycles inK2mPlus Perfect Matching

We1999 Academic Pressanalyze the freedom one has when constructing an Euler cycle throughCopyright K2m+, the complete graph of even order plus a perfect matching. We start with the fact that a multigraphGwithmedges is Eulerian iff it can be obtained from a cycle of lengthmby glueing certain vertices together. Our extension is the following. We deal with cycles of (2m2)+medges, whose vertices are colored black and white. There are 2mblack vertices and (2m2)?mwhite vertices. Assuming that glueing a black and a white vertex results in a black vertex we describe a method of how to glue certain vertices of a given cycle in such a way that the resulting multigraph isK2m+with all vertices black.

Euler cycles in the complete graph K 2 m +1

Abstract We analyze the freedom one has when walking along an Euler cycle through a complete graph of an odd order: Is it possible, for any cycle C of ( 2 2 m + 1 ) vertices, 2 m + 1 of them being black, to find an edge monomorphism of C onto K 2 m + 1 , that would be injective on the set of black vertices of C ? It is shown that the answer is positive for all but two cases. Our proof is constructive, however, we relied on computers to verify approximately 37 000 cases needed for the induction basis. Our theorem generalizes a previous result on the decomposition of K 2 m + 1 into edge-disjoint trails of given lengths. In addition, a relation to the concept of harmonious chromatic number is mentioned.