245 research outputs found
On the trace of random walks on random graphs
We study graph-theoretic properties of the trace of a random walk on a random
graph. We show that for any there exists such that the
trace of the simple random walk of length on the
random graph for is, with high probability,
Hamiltonian and -connected. In the special case (i.e.
when ), we show a hitting time result according to which, with high
probability, exactly one step after the last vertex has been visited, the trace
becomes Hamiltonian, and one step after the last vertex has been visited for
the 'th time, the trace becomes -connected.Comment: 32 pages, revised versio
Packing a bin online to maximize the total number of items
A bin of capacity 1 and a nite sequence of items of\ud
sizes a1; a2; : : : are considered, where the items are given one by one\ud
without information about the future. An online algorithm A must\ud
irrevocably decide whether or not to put an item into the bin whenever\ud
it is presented. The goal is to maximize the number of items collected.\ud
A is f-competitive for some function f if n() f(nA()) holds for all\ud
sequences , where n is the (theoretical) optimum and nA the number\ud
of items collected by A.\ud
A necessary condition on f for the existence of an f-competitive\ud
(possibly randomized) online algorithm is given. On the other hand,\ud
this condition is seen to guarantee the existence of a deterministic online\ud
algorithm that is "almost" f-competitive in a well-dened sense
A Survey of Best Monotone Degree Conditions for Graph Properties
We survey sufficient degree conditions, for a variety of graph properties,
that are best possible in the same sense that Chvatal's well-known degree
condition for hamiltonicity is best possible.Comment: 25 page
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
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