2,272 research outputs found
Spanning Trees in Graphs of High Minimum Degree with a Universal Vertex I: An Approximate Asymptotic Result
In this paper and a companion paper, we prove that, if is sufficiently
large, every graph on vertices that has a universal vertex and minimum
degree at least contains each tree with
edges as a subgraph. The present paper already contains an approximate
asymptotic version of the result.
Our result confirms, for large , an important special case of a recent
conjecture by Havet, Reed, Stein, and Wood.Comment: 46 page
Spanning Trees in Graphs of High Minimum Degree which have a Universal Vertex II: A Tight Result
We prove that, if is sufficiently large, every graph on vertices
that has a universal vertex and minimum degree at least contains each tree with edges as a subgraph. Our result
confirms, for large , an important special case of a conjecture by Havet,
Reed, Stein, and Wood.
The present paper builds on the results of a companion paper in which we
proved the statement for all trees having a vertex that is adjacent to many
leaves.Comment: 29 page
Minima in branching random walks
Given a branching random walk, let be the minimum position of any
member of the th generation. We calculate to within O(1) and
prove exponential tail bounds for , under
quite general conditions on the branching random walk. In particular, together
with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89--108], our results
fully characterize the possible behavior of when the branching
random walk has bounded branching and step size.Comment: Published in at http://dx.doi.org/10.1214/08-AOP428 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Forcing a sparse minor
This paper addresses the following question for a given graph : what is
the minimum number such that every graph with average degree at least
contains as a minor? Due to connections with Hadwiger's Conjecture,
this question has been studied in depth when is a complete graph. Kostochka
and Thomason independently proved that . More generally,
Myers and Thomason determined when has a super-linear number of
edges. We focus on the case when has a linear number of edges. Our main
result, which complements the result of Myers and Thomason, states that if
has vertices and average degree at least some absolute constant, then
. Furthermore, motivated by the case when
has small average degree, we prove that if has vertices and edges,
then (where the coefficient of 1 in the term is best
possible)
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