905 research outputs found
On covering expander graphs by Hamilton cycles
The problem of packing Hamilton cycles in random and pseudorandom graphs has
been studied extensively. In this paper, we look at the dual question of
covering all edges of a graph by Hamilton cycles and prove that if a graph with
maximum degree satisfies some basic expansion properties and contains
a family of edge disjoint Hamilton cycles, then there also
exists a covering of its edges by Hamilton cycles. This
implies that for every and every there exists
a covering of all edges of by Hamilton cycles
asymptotically almost surely, which is nearly optimal.Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some
math/061275
Degree-3 Treewidth Sparsifiers
We study treewidth sparsifiers. Informally, given a graph of treewidth
, a treewidth sparsifier is a minor of , whose treewidth is close to
, is small, and the maximum vertex degree in is bounded.
Treewidth sparsifiers of degree are of particular interest, as routing on
node-disjoint paths, and computing minors seems easier in sub-cubic graphs than
in general graphs.
In this paper we describe an algorithm that, given a graph of treewidth
, computes a topological minor of such that (i) the treewidth of
is ; (ii) ; and (iii) the maximum
vertex degree in is . The running time of the algorithm is polynomial in
and . Our result is in contrast to the known fact that unless , treewidth does not admit polynomial-size kernels.
One of our key technical tools, which is of independent interest, is a
construction of a small minor that preserves node-disjoint routability between
two pairs of vertex subsets. This is closely related to the open question of
computing small good-quality vertex-cut sparsifiers that are also minors of the
original graph.Comment: Extended abstract to appear in Proceedings of ACM-SIAM SODA 201
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
Logarithmically-small Minors and Topological Minors
Mader proved that for every integer there is a smallest real number
such that any graph with average degree at least must contain a
-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with
vertices and average degree at least must contain a -minor
consisting of at most vertices. Shapira and Sudakov
subsequently proved that such a graph contains a -minor consisting of at
most vertices. Here we build on their method
using graph expansion to remove the factor and prove the
conjecture.
Mader also proved that for every integer there is a smallest real number
such that any graph with average degree larger than must contain
a -topological minor. We prove that, for sufficiently large , graphs
with average degree at least contain a -topological
minor consisting of at most vertices. Finally, we show
that, for sufficiently large , graphs with average degree at least
contain either a -minor consisting of at most
vertices or a -topological minor consisting of at most
vertices.Comment: 19 page
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
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