2,603 research outputs found
Looking for vertex number one
Given an instance of the preferential attachment graph , we
would like to find vertex 1, using only 'local' information about the graph;
that is, by exploring the neighborhoods of small sets of vertices. Borgs et. al
gave an an algorithm which runs in time , which is local in the
sense that at each step, it needs only to search the neighborhood of a set of
vertices of size . We give an algorithm to find vertex 1, which
w.h.p. runs in time and which is local in the strongest sense
of operating only on neighborhoods of single vertices. Here
is any function that goes to infinity with .Comment: As accepted for AA
The topology of competitively constructed graphs
We consider a simple game, the -regular graph game, in which players take
turns adding edges to an initially empty graph subject to the constraint that
the degrees of vertices cannot exceed . We show a sharp topological
threshold for this game: for the case a player can ensure the resulting
graph is planar, while for the case , a player can force the appearance of
arbitrarily large clique minors.Comment: 9 pages, 2 figure
Traveling in randomly embedded random graphs
We consider the problem of traveling among random points in Euclidean space,
when only a random fraction of the pairs are joined by traversable connections.
In particular, we show a threshold for a pair of points to be connected by a
geodesic of length arbitrarily close to their Euclidean distance, and analyze
the minimum length Traveling Salesperson Tour, extending the
Beardwood-Halton-Hammersley theorem to this setting.Comment: 25 pages, 2 figure
Between 2- and 3-colorability
We consider the question of the existence of homomorphisms between
and odd cycles when . We show that for any positive integer
, there exists such that if then
w.h.p. has a homomorphism from to so long as
its odd-girth is at least . On the other hand, we show that if
then w.h.p. there is no homomorphism from to . Note that in our
range of interest, w.h.p., implying that there is a
homomorphism from to
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