1,952 research outputs found
Two Problems on Bipartite Graphs
Erdos proved the well-known result that every graph has a spanning, bipartite subgraph such that every vertex has degree at least half of its original degree. Bollobas and Scott conjectured that one can get a slightly weaker result if we require the subgraph to be not only spanning and bipartite, but also balanced. We prove this conjecture for graphs of maximum degree 3.
The majority of the paper however, will focus on graph tiling. Graph tiling (or sometimes referred to as graph packing) is where, given a graph H, we find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove an asymptotic minimum degree condition for an arbitrary bipartite graph G to be tiled by another arbitrary bipartite graph H. This proves a conjecture of Zhao and also implies an asymptotic version of a result of Kuhn and Osthus for bipartite graphs
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
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