1,213 research outputs found
Improved Bounds for the Graham-Pollak Problem for Hypergraphs
For a fixed , let denote the minimum number of complete
-partite -graphs needed to partition the complete -graph on
vertices. The Graham-Pollak theorem asserts that . An easy
construction shows that ,
and we write for the least number such that .
It was known that for each even , but this was not known
for any odd value of . In this short note, we prove that . Our
method also shows that , answering another open problem
Transversal designs and induced decompositions of graphs
We prove that for every complete multipartite graph there exist very
dense graphs on vertices, namely with as many as
edges for all , for some constant , such that can be
decomposed into edge-disjoint induced subgraphs isomorphic to~. This result
identifies and structurally explains a gap between the growth rates and
on the minimum number of non-edges in graphs admitting an
induced -decomposition
Completion and deficiency problems
Given a partial Steiner triple system (STS) of order , what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order with at
most triples, it can always be embedded into a complete
STS of order , which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property and a graph , we define the
deficiency of the graph with respect to the property to be
the smallest positive integer such that the join has property
. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a -decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given
Resolution of the Oberwolfach problem
The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of
into edge-disjoint copies of a given -factor. We show that this
can be achieved for all large . We actually prove a significantly more
general result, which allows for decompositions into more general types of
factors. In particular, this also resolves the Hamilton-Waterloo problem for
large .Comment: 28 page
A limit law of almost -partite graphs
For integers , we study (undirected) graphs with
vertices such that the vertices can be partitioned into parts
such that every vertex has at most neighbours in its own part. The set of
all such graphs is denoted \mbP_n(l,d). We prove a labelled first-order limit
law, i.e., for every first-order sentence , the proportion of graphs
in \mbP_n(l,d) that satisfy converges as . By
combining this result with a result of Hundack, Pr\"omel and Steger \cite{HPS}
we also prove that if are integers, then
\mb{Forb}(\mcK_{1, s_1, ..., s_l}) has a labelled first-order limit law,
where \mb{Forb}(\mcK_{1, s_1, ..., s_l}) denotes the set of all graphs with
vertices , for some , in which there is no subgraph isomorphic to
the complete -partite graph with parts of sizes . In
the course of doing this we also prove that there exists a first-order formula
(depending only on and ) such that the proportion of \mcG \in
\mbP_n(l,d) with the following property approaches 1 as : there
is a unique partition of into parts such that every vertex
has at most neighbours in its own part, and this partition, viewed as an
equivalence relation, is defined by
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
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