1,826 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
Quadri-tilings of the plane
We introduce {\em quadri-tilings} and show that they are in bijection with
dimer models on a {\em family} of graphs arising from rhombus
tilings. Using two height functions, we interpret a sub-family of all
quadri-tilings, called {\em triangular quadri-tilings}, as an interface model
in dimension 2+2. Assigning "critical" weights to edges of , we prove an
explicit expression, only depending on the local geometry of the graph ,
for the minimal free energy per fundamental domain Gibbs measure; this solves a
conjecture of \cite{Kenyon1}. We also show that when edges of are
asymptotically far apart, the probability of their occurrence only depends on
this set of edges. Finally, we give an expression for a Gibbs measure on the
set of {\em all} triangular quadri-tilings whose marginals are the above Gibbs
measures, and conjecture it to be that of minimal free energy per fundamental
domain.Comment: Revised version, minor changes. 30 pages, 13 figure
Asymptotic multipartite version of the Alon-Yuster theorem
In this paper, we prove the asymptotic multipartite version of the
Alon-Yuster theorem, which is a generalization of the Hajnal-Szemer\'edi
theorem: If is an integer, is a -colorable graph and
is fixed, then, for every sufficiently large , where
divides , and for every balanced -partite graph on vertices with
each of its corresponding bipartite subgraphs having minimum
degree at least , has a subgraph consisting of
vertex-disjoint copies of .
The proof uses the Regularity method together with linear programming.Comment: 22 pages, 1 figur
Ramsey numbers of cubes versus cliques
The cube graph Q_n is the skeleton of the n-dimensional cube. It is an
n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N
such that every graph of order N contains the cube graph Q_n or an independent
set of order s. Burr and Erdos in 1983 asked whether the simple lower bound
r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large.
We make progress on this problem, obtaining the first upper bound which is
within a constant factor of the lower bound.Comment: 26 page
Tilings in randomly perturbed dense graphs
A perfect -tiling in a graph is a collection of vertex-disjoint copies
of a graph in that together cover all the vertices in . In this
paper we investigate perfect -tilings in a random graph model introduced by
Bohman, Frieze and Martin in which one starts with a dense graph and then adds
random edges to it. Specifically, for any fixed graph , we determine the
number of random edges required to add to an arbitrary graph of linear minimum
degree in order to ensure the resulting graph contains a perfect -tiling
with high probability. Our proof utilises Szemer\'edi's Regularity lemma as
well as a special case of a result of Koml\'os concerning almost perfect
-tilings in dense graphs.Comment: 19 pages, to appear in CP
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