63,097 research outputs found
Cycle packing
In the 1960s, Erd\H{o}s and Gallai conjectured that the edge set of every
graph on n vertices can be partitioned into O(n) cycles and edges. They
observed that one can easily get an O(n log n) upper bound by repeatedly
removing the edges of the longest cycle. We make the first progress on this
problem, showing that O(n log log n) cycles and edges suffice. We also prove
the Erd\H{o}s-Gallai conjecture for random graphs and for graphs with linear
minimum degree.Comment: 18 page
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
Finding long cycles in graphs
We analyze the problem of discovering long cycles inside a graph. We propose
and test two algorithms for this task. The first one is based on recent
advances in statistical mechanics and relies on a message passing procedure.
The second follows a more standard Monte Carlo Markov Chain strategy. Special
attention is devoted to Hamiltonian cycles of (non-regular) random graphs of
minimal connectivity equal to three
Edge-disjoint Hamilton cycles in graphs
In this paper we give an approximate answer to a question of Nash-Williams
from 1970: we show that for every \alpha > 0, every sufficiently large graph on
n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8
edge-disjoint Hamilton cycles. More generally, we give an asymptotically best
possible answer for the number of edge-disjoint Hamilton cycles that a graph G
with minimum degree \delta must have. We also prove an approximate version of
another long-standing conjecture of Nash-Williams: we show that for every
\alpha > 0, every (almost) regular and sufficiently large graph on n vertices
with minimum degree at least can be almost decomposed into
edge-disjoint Hamilton cycles.Comment: Minor Revisio
Hamilton decompositions of regular tournaments
We show that every sufficiently large regular tournament can almost
completely be decomposed into edge-disjoint Hamilton cycles. More precisely,
for each \eta>0 every regular tournament G of sufficiently large order n
contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an
approximate solution to a conjecture of Kelly from 1968. Our result also
extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our
main result. To appear in the Proceedings of the LM
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