2,176 research outputs found
Sparse Kneser graphs are Hamiltonian
For integers and , the Kneser graph is the
graph whose vertices are the -element subsets of and whose
edges connect pairs of subsets that are disjoint. The Kneser graphs of the form
are also known as the odd graphs. We settle an old problem due to
Meredith, Lloyd, and Biggs from the 1970s, proving that for every ,
the odd graph has a Hamilton cycle. This and a known conditional
result due to Johnson imply that all Kneser graphs of the form
with and have a Hamilton cycle. We also prove that
has at least distinct Hamilton cycles for .
Our proofs are based on a reduction of the Hamiltonicity problem in the odd
graph to the problem of finding a spanning tree in a suitably defined
hypergraph on Dyck words
Proof of Koml\'os's conjecture on Hamiltonian subsets
Koml\'os conjectured in 1981 that among all graphs with minimum degree at
least , the complete graph minimises the number of Hamiltonian
subsets, where a subset of vertices is Hamiltonian if it contains a spanning
cycle. We prove this conjecture when is sufficiently large. In fact we
prove a stronger result: for large , any graph with average degree at
least contains almost twice as many Hamiltonian subsets as ,
unless is isomorphic to or a certain other graph which we
specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ
Hamilton cycles in highly connected and expanding graphs
In this paper we prove a sufficient condition for the existence of a Hamilton
cycle, which is applicable to a wide variety of graphs, including relatively
sparse graphs. In contrast to previous criteria, ours is based on only two
properties: one requiring expansion of ``small'' sets, the other ensuring the
existence of an edge between any two disjoint ``large'' sets. We also discuss
applications in positional games, random graphs and extremal graph theory.Comment: 19 page
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
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
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