205 research outputs found
The extremal function for partial bipartite tilings
For a fixed bipartite graph H and given number c, 0<c<1, we determine the
threshold T_H(c) which guarantees that any n-vertex graph with at edge density
at least T_H(c) contains vertex-disjoint copies of H. In the
proof we use a variant of a technique developed by
Komlos~\bcolor{[Combinatorica 20 (2000), 203-218}]Comment: 10 page
A density Corr\'adi-Hajnal Theorem
We find, for all sufficiently large and each , the maximum number of
edges in an -vertex graph which does not contain vertex-disjoint
triangles.
This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in
turn an extension of Mantel's Theorem. Our result can also be viewed as a
density version of the Corradi-Hajnal Theorem.Comment: 41 pages (including 11 pages of appendix), 4 figures, 2 table
On the Existence of Loose Cycle Tilings and Rainbow Cycles
abstract: Extremal graph theory results often provide minimum degree
conditions which guarantee a copy of one graph exists within
another. A perfect -tiling of a graph is a collection
of subgraphs of such that every element of
is isomorphic to and such that every vertex in
is in exactly one element of . Let denote
the loose cycle on vertices, the -uniform hypergraph
obtained by replacing the edges of a graph cycle
on vertices with edge triples , where is
uniquely assigned to . This dissertation proves for even
, that any sufficiently large -uniform hypergraph
on vertices with minimum -degree
\delta^1(H) \geq {n - 1 \choose 2} - {\Bsize \choose 2} + c(t,n) +
1, where , contains a perfect
-tiling. The result is tight, generalizing previous
results on by Han and Zhao. For an edge colored graph ,
let the minimum color degree be the minimum number of
distinctly colored edges incident to a vertex. Call rainbow if
every edge has a unique color. For , this dissertation
proves that any sufficiently large edge colored graph on
vertices with contains a rainbow
cycle on vertices. The result is tight for odd and
extends previous results for . In addition, for even
, this dissertation proves that any sufficiently large
edge colored graph on vertices with
, where
, contains a rainbow cycle on
vertices. The result is tight when . As a related
result, this dissertation proves for all , that any
sufficiently large oriented graph on vertices with
contains a directed cycle on
vertices. This partially generalizes a result by Kelly,
K\"uhn, and Osthus that uses minimum semidegree rather than minimum
out degree.Dissertation/ThesisDoctoral Dissertation Mathematics 201
How quickly can we sample a uniform domino tiling of the 2L x 2L square via Glauber dynamics?
TThe prototypical problem we study here is the following. Given a square, there are approximately ways to tile it with
dominos, i.e. with horizontal or vertical rectangles, where
is Catalan's constant [Kasteleyn '61, Temperley-Fisher '61]. A
conceptually simple (even if computationally not the most efficient) way of
sampling uniformly one among so many tilings is to introduce a Markov Chain
algorithm (Glauber dynamics) where, with rate , two adjacent horizontal
dominos are flipped to vertical dominos, or vice-versa. The unique invariant
measure is the uniform one and a classical question [Wilson
2004,Luby-Randall-Sinclair 2001] is to estimate the time it takes to
approach equilibrium (i.e. the running time of the algorithm). In
[Luby-Randall-Sinclair 2001, Randall-Tetali 2000], fast mixin was proven:
for some finite . Here, we go much beyond and show that . Our result applies to rather general domain
shapes (not just the square), provided that the typical height
function associated to the tiling is macroscopically planar in the large
limit, under the uniform measure (this is the case for instance for the
Temperley-type boundary conditions considered in [Kenyon 2000]). Also, our
method extends to some other types of tilings of the plane, for instance the
tilings associated to dimer coverings of the hexagon or square-hexagon
lattices.Comment: to appear on PTRF; 42 pages, 9 figures; v2: typos corrected,
references adde
A (2+1)-dimensional growth process with explicit stationary measures
We introduce a class of (2+1)-dimensional stochastic growth processes, that
can be seen as irreversible random dynamics of discrete interfaces.
"Irreversible" means that the interface has an average non-zero drift.
Interface configurations correspond to height functions of dimer coverings of
the infinite hexagonal or square lattice. The model can also be viewed as an
interacting driven particle system and in the totally asymmetric case the
dynamics corresponds to an infinite collection of mutually interacting
Hammersley processes.
When the dynamical asymmetry parameter equals zero, the
infinite-volume Gibbs measures (with given slope ) are
stationary and reversible. When , are not reversible any
more but, remarkably, they are still stationary. In such stationary states, we
find that the average height function at any given point grows linearly
with time with a non-zero speed: while the typical fluctuations of are
smaller than any power of as .
In the totally asymmetric case of and on the hexagonal lattice, the
dynamics coincides with the "anisotropic KPZ growth model" introduced by A.
Borodin and P. L. Ferrari. For a suitably chosen, "integrable", initial
condition (that is very far from the stationary state), they were able to
determine the hydrodynamic limit and a CLT for interface fluctuations on scale
, exploiting the fact that in that case certain space-time
height correlations can be computed exactly.Comment: 37 pages, 13 figures. v3: some references added, introduction
expanded, minor changes in the bul
Matchings in 3-uniform hypergraphs
We determine the minimum vertex degree that ensures a perfect matching in a
3-uniform hypergraph. More precisely, suppose that H is a sufficiently large
3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex
degree of H is greater than \binom{n-1}{2}-\binom{2n/3}{2}, then H contains a
perfect matching. This bound is tight and answers a question of Han, Person and
Schacht. More generally, we show that H contains a matching of size d\le n/3 if
its minimum vertex degree is greater than \binom{n-1}{2}-\binom{n-d}{2}, which
is also best possible. This extends a result of Bollobas, Daykin and Erdos.Comment: 18 pages, 1 figure. To appear in JCT
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