1,545 research outputs found
Internal Partitions of Regular Graphs
An internal partition of an -vertex graph is a partition of
such that every vertex has at least as many neighbors in its own part as in the
other part. It has been conjectured that every -regular graph with
vertices has an internal partition. Here we prove this for . The case
is of particular interest and leads to interesting new open problems on
cubic graphs. We also provide new lower bounds on and find new families
of graphs with no internal partitions. Weighted versions of these problems are
considered as well
On the number of 4-cycles in a tournament
If is an -vertex tournament with a given number of -cycles, what
can be said about the number of its -cycles? The most interesting range of
this problem is where is assumed to have cyclic triples for
some and we seek to minimize the number of -cycles. We conjecture that
the (asymptotic) minimizing is a random blow-up of a constant-sized
transitive tournament. Using the method of flag algebras, we derive a lower
bound that almost matches the conjectured value. We are able to answer the
easier problem of maximizing the number of -cycles. These questions can be
equivalently stated in terms of transitive subtournaments. Namely, given the
number of transitive triples in , how many transitive quadruples can it
have? As far as we know, this is the first study of inducibility in
tournaments.Comment: 11 pages, 5 figure
Monotone Maps, Sphericity and Bounded Second Eigenvalue
We consider {\em monotone} embeddings of a finite metric space into low
dimensional normed space. That is, embeddings that respect the order among the
distances in the original space. Our main interest is in embeddings into
Euclidean spaces. We observe that any metric on points can be embedded into
, while, (in a sense to be made precise later), for almost every
-point metric space, every monotone map must be into a space of dimension
.
It becomes natural, then, to seek explicit constructions of metric spaces
that cannot be monotonically embedded into spaces of sublinear dimension. To
this end, we employ known results on {\em sphericity} of graphs, which suggest
one example of such a metric space - that defined by a complete bipartitegraph.
We prove that an -regular graph of order , with bounded diameter
has sphericity , where is the second
largest eigenvalue of the adjacency matrix of the graph, and 0 < \delta \leq
\half is constant. We also show that while random graphs have linear
sphericity, there are {\em quasi-random} graphs of logarithmic sphericity.
For the above bound to be linear, must be constant. We show that
if the second eigenvalue of an -regular graph is bounded by a constant,
then the graph is close to being complete bipartite. Namely, its adjacency
matrix differs from that of a complete bipartite graph in only
entries. Furthermore, for any 0 < \delta < \half, and , there are
only finitely many -regular graphs with second eigenvalue at most
Graphs with few 3-cliques and 3-anticliques are 3-universal
For given integers k, l we ask whether every large graph with a sufficiently
small number of k-cliques and k-anticliques must contain an induced copy of
every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A
similar phenomenon is established as well for tournaments with k=l=4.Comment: 12 pages, 1 figur
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