18 research outputs found
Counting Hamilton cycles in sparse random directed graphs
Let D(n,p) be the random directed graph on n vertices where each of the
n(n-1) possible arcs is present independently with probability p. A celebrated
result of Frieze shows that if then D(n,p) typically
has a directed Hamilton cycle, and this is best possible. In this paper, we
obtain a strengthening of this result, showing that under the same condition,
the number of directed Hamilton cycles in D(n,p) is typically
. We also prove a hitting-time version of this statement,
showing that in the random directed graph process, as soon as every vertex has
in-/out-degrees at least 1, there are typically
directed Hamilton cycles
The number of Hamiltonian decompositions of regular graphs
A Hamilton cycle in a graph is a cycle passing through every vertex
of . A Hamiltonian decomposition of is a partition of its edge
set into disjoint Hamilton cycles. One of the oldest results in graph theory is
Walecki's theorem from the 19th century, showing that a complete graph on
an odd number of vertices has a Hamiltonian decomposition. This result was
recently greatly extended by K\"{u}hn and Osthus. They proved that every
-regular -vertex graph with even degree for some fixed
has a Hamiltonian decomposition, provided is sufficiently
large. In this paper we address the natural question of estimating ,
the number of such decompositions of . Our main result is that
. In particular, the number of Hamiltonian
decompositions of is
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio