28 research outputs found
Sharp Concentration of Hitting Size for Random Set Systems
Consider the random set system of {1,2,...,n}, where each subset in the power
set is chosen independently with probability p. A set H is said to be a hitting
set if it intersects each chosen set. The second moment method is used to
exhibit the sharp concentration of the minimal size of H for a variety of
values of p.Comment: 11 page
Conflict-free coloring of graphs
We study the conflict-free chromatic number chi_{CF} of graphs from extremal
and probabilistic point of view. We resolve a question of Pach and Tardos about
the maximum conflict-free chromatic number an n-vertex graph can have. Our
construction is randomized. In relation to this we study the evolution of the
conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and
give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the
conflict-free chromatic number differs from the domination number by at most 3.Comment: 12 page
Best and worst case permutations for random online domination of the path
We study a randomized algorithm for graph domination, by which, according to
a uniformly chosen permutation, vertices are revealed and added to the
dominating set if not already dominated. We determine the expected size of the
dominating set produced by the algorithm for the path graph and use this
to derive the expected size for some related families of graphs. We then
provide a much-refined analysis of the worst and best cases of this algorithm
on and enumerate the permutations for which the algorithm has the
worst-possible performance and best-possible performance. The case of
dominating the path graph has connections to previous work of Bouwer and Star,
and of Gessel on greedily coloring the path.Comment: 13 pages, 1 figur
The bondage number of random graphs
A dominating set of a graph is a subset of its vertices such that every
vertex not in is adjacent to at least one member of . The domination
number of a graph is the number of vertices in a smallest dominating set of
. The bondage number of a nonempty graph is the size of a smallest set
of edges whose removal from results in a graph with domination number
greater than the domination number of . In this note, we study the bondage
number of binomial random graph . We obtain a lower bound that matches
the order of the trivial upper bound. As a side product, we give a one-point
concentration result for the domination number of under certain
restrictions