1,567 research outputs found
Towards Resistance Sparsifiers
We study resistance sparsification of graphs, in which the goal is to find a
sparse subgraph (with reweighted edges) that approximately preserves the
effective resistances between every pair of nodes. We show that every dense
regular expander admits a -resistance sparsifier of size , and conjecture this bound holds for all graphs on nodes. In
comparison, spectral sparsification is a strictly stronger notion and requires
edges even on the complete graph.
Our approach leads to the following structural question on graphs: Does every
dense regular expander contain a sparse regular expander as a subgraph? Our
main technical contribution, which may of independent interest, is a positive
answer to this question in a certain setting of parameters. Combining this with
a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the
aforementioned resistance sparsifiers
Percolation on sparse random graphs with given degree sequence
We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.Comment: 20 page
Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions
In a sequence of four papers, we prove the following results (via a unified
approach) for all sufficiently large :
(i) [1-factorization conjecture] Suppose that is even and . Then every -regular graph on vertices has a
decomposition into perfect matchings. Equivalently, .
(ii) [Hamilton decomposition conjecture] Suppose that . Then every -regular graph on vertices has a decomposition
into Hamilton cycles and at most one perfect matching.
(iii) We prove an optimal result on the number of edge-disjoint Hamilton
cycles in a graph of given minimum degree.
According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer
questions of Nash-Williams from 1970. The above bounds are best possible. In
the current paper, we show the following: suppose that is close to a
complete balanced bipartite graph or to the union of two cliques of equal size.
If we are given a suitable set of path systems which cover a set of
`exceptional' vertices and edges of , then we can extend these path systems
into an approximate decomposition of into Hamilton cycles (or perfect
matchings if appropriate).Comment: We originally split the proof into four papers, of which this was the
third paper. We have now combined this series into a single publication
[arXiv:1401.4159v2], which will appear in the Memoirs of the AMS. 29 pages, 2
figure
A Geometric Theory for Hypergraph Matching
We develop a theory for the existence of perfect matchings in hypergraphs
under quite general conditions. Informally speaking, the obstructions to
perfect matchings are geometric, and are of two distinct types: 'space
barriers' from convex geometry, and 'divisibility barriers' from arithmetic
lattice-based constructions. To formulate precise results, we introduce the
setting of simplicial complexes with minimum degree sequences, which is a
generalisation of the usual minimum degree condition. We determine the
essentially best possible minimum degree sequence for finding an almost perfect
matching. Furthermore, our main result establishes the stability property:
under the same degree assumption, if there is no perfect matching then there
must be a space or divisibility barrier. This allows the use of the stability
method in proving exact results. Besides recovering previous results, we apply
our theory to the solution of two open problems on hypergraph packings: the
minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's
conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we
prove the exact result for tetrahedra and the asymptotic result for Fischer's
conjecture; since the exact result for the latter is technical we defer it to a
subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical
Society. 101 pages. v2: minor changes including some additional diagrams and
passages of expository tex
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