81 research outputs found
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
Dirac's theorem for random regular graphs
We prove a `resilience' version of Dirac's theorem in the setting of random
regular graphs. More precisely, we show that, whenever is sufficiently
large compared to , a.a.s. the following holds: let be any
subgraph of the random -vertex -regular graph with minimum
degree at least . Then is Hamiltonian.
This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result
is best possible: firstly, the condition that is large cannot be omitted,
and secondly, the minimum degree bound cannot be improved.Comment: Final accepted version, to appear in Combinatorics, Probability &
Computin
On sufficient conditions for Hamiltonicity in dense graphs
We study structural conditions in dense graphs that guarantee the existence
of vertex-spanning substructures such as Hamilton cycles. It is easy to see
that every Hamiltonian graph is connected, has a perfect fractional matching
and, excluding the bipartite case, contains an odd cycle. Our main result in
turn states that any large enough graph that robustly satisfies these
properties must already be Hamiltonian. Moreover, the same holds for embedding
powers of cycles and graphs of sublinear bandwidth subject to natural
generalisations of connectivity, matchings and odd cycles.
This solves the embedding problem that underlies multiple lines of research
on sufficient conditions for Hamiltonicity in dense graphs. As applications, we
recover and establish Bandwidth Theorems in a variety of settings including
Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type
conditions, locally dense and inseparable graphs, multipartite graphs as well
as robust expanders
Robust Hamiltonicity in families of Dirac graphs
A graph is called Dirac if its minimum degree is at least half of the number
of vertices in it. Joos and Kim showed that every collection
of Dirac graphs on the same vertex set of
size contains a Hamilton cycle transversal, i.e., a Hamilton cycle on
with a bijection such that
for every .
In this paper, we determine up to a multiplicative constant, the threshold
for the existence of a Hamilton cycle transversal in a collection of random
subgraphs of Dirac graphs in various settings. Our proofs rely on constructing
a spread measure on the set of Hamilton cycle transversals of a family of Dirac
graphs.
As a corollary, we obtain that every collection of Dirac graphs on
vertices contains at least different Hamilton cycle transversals
for some absolute constant . This is optimal up to the constant
. Finally, we show that if is sufficiently large, then every such
collection spans pairwise edge-disjoint Hamilton cycle transversals, and
this is best possible. These statements generalize classical counting results
of Hamilton cycles in a single Dirac graph
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