Tomasz Luczak

A Parallel Randomized Algorithm for Finding a Maximal Independent Set in a Linear Hypergraph

We present a randomized parallel algorithm with polylogarithmic expected running time for finding a maximal independent set in a linear hypergraph.

Stability of vertices in random boolean cellular automata

Based on computer simulations, Kauffman (Physica D, 10, 145-156, 1984) made several generalizations about a random Boolean cellular automaton which he invented as a model of cellular metabolism. Here we give the first rigorous proofs of two of Kauffmans generalizations: a large fraction of vertices stabilize quickly, consequently the length of cycles in the automatons behavior is small compared to that of a random mapping with the same number of states; and reversal of the states of a large fraction of the vertices does not affect the cycle to which the automaton moves.

On induced Ramsey numbers

The induced Ramsey number IR(G,H) is defined as the smallest integer n, for which there exists a graph F on n vertices such that any 2-colouring of its edges with red and blue leads to either a red copy of G induced in F, or an induced blue H. In this note, we study the value of the induced Ramsey numbers, as well as their planar and weak versions, for some special classes of graphs. In particular, we show that, for the induced planar Ramsey numbers, the fact whether we prohibit monochromatic copies induced in the graph, or induced just in its own colour, may significantly affect the value of the Ramsey number.

Maximal induced trees in sparse random graphs

A study of the orders of maximal induced trees in a random graph G p with small edge probability p is given. In particular, it is shown that the giant component of almost every G p , where p = c/n and c > 1 is a constant, contains only very small maximal trees (that are of a specific type) and very large maximal trees. The presented results provide an elementary proof of a conjecture from [3] that was confirmed recently in [4] and [5].

A Greedy Algorithm Estimating the Height of Random Trees

The behavior of a greedy algorithm which estimates the height of a random, labelled rooted tree is studied. A self-similarity argument is used to characterize the limit distribution of the length H of the path found by such an algorithm in a random rooted tree as the unique solution of an integral equation. Furthermore, it is shown that

Ramsey properties of families of graphs

\lim_{n\rightarrow\infty}\frac{{\rm{E}} H}{\sqrt n} =\frac{\sqrt{2\pi}}{2\sqrt 2-{\rm{ln}} (3+2\sqrt 2)} = 2.352139...,

Random Graphs

i.e., the expected length of the path constructed by the algorithm is roughly 93.8 of the expected height of a random rooted tree.

Bootstrap percolation on the random graph Gn,p

We investigate a problem of maintaining a target population of mobile agents in a distributed system. The purpose of the agents is to perform certain activities, so the goal is to avoid overpopulation (leading to waste of resources) as well as underpopulation (resulting in a poor service). We assume that there must be no centralized control over the number of agents, since it might result in systems vulnerability. We analyze a simple protocol in which each node keeps at most one copy of an agent and if there is a single agent in a node, a new agent is born with a certain probability p. At each time step the agents migrate independently at random to chosen locations. We show that during a protocol execution the number of agents stabilizes around a level depending on p. We derive analytically simple formulas that determine probability p based on the target fraction of nodes holding an agent. The previous proposals of this type were based on experimental data only.