55 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 Lefthanded Local Lemma characterizes chordal dependency graphs
Shearer gave a general theorem characterizing the family \LLL of dependency
graphs labeled with probabilities which have the property that for any
family of events with a dependency graph from \LLL (whose vertex-labels are
upper bounds on the probabilities of the events), there is a positive
probability that none of the events from the family occur.
We show that, unlike the standard Lov\'asz Local Lemma---which is less
powerful than Shearer's condition on every nonempty graph---a recently proved
`Lefthanded' version of the Local Lemma is equivalent to Shearer's condition
for all chordal graphs. This also leads to a simple and efficient algorithm to
check whether a given labeled chordal graph is in \LLL.Comment: 12 pages, 1 figur
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
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
A note on dispersing particles on a line
We consider a synchronous dispersion process introduced in \cite{CRRS} and we
show that on the infinite line the final set of occupied sites takes up
space, where is the number of particles involved
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