881 research outputs found
An extremal theorem in the hypercube
The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two
vertices are adjacent if they differ in exactly one coordinate. For any
subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a
subgraph of Q_n which does not contain a copy of H. We find a wide class of
subgraphs H, including all previously known examples, for which ex(Q_n, H) =
o(e(Q_n)). In particular, our method gives a unified approach to proving that
ex(Q_n, C_{2t}) = o(e(Q_n)) for all t >= 4 other than 5.Comment: 6 page
Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube
In this paper we modify slightly Razborov's flag algebra machinery to be
suitable for the hypercube. We use this modified method to show that the
maximum number of edges of a 4-cycle-free subgraph of the n-dimensional
hypercube is at most 0.6068 times the number of its edges. We also improve the
upper bound on the number of edges for 6-cycle-free subgraphs of the
n-dimensional hypercube from the square root of 2 - 1 to 0.3755 times the
number of its edges. Additionally, we show that if the n-dimensional hypercube
is considered as a poset, then the maximum vertex density of three middle
layers in an induced subgraph without 4-cycles is at most 2.15121 times n
choose n/2.Comment: 14 pages, 9 figure
Problems in extremal graph theory
We consider a variety of problems in extremal graph and set theory.
The {\em chromatic number} of , , is the smallest integer
such that is -colorable.
The {\it square} of , written , is the supergraph of in which also
vertices within distance 2 of each other in are adjacent.
A graph is a {\it minor} of if
can be obtained from a subgraph of by contracting edges.
We show that the upper bound for
conjectured by Wegner (1977) for planar graphs
holds when is a -minor-free graph.
We also show that is equal to the bound
only when contains a complete graph of that order.
One of the central problems of extremal hypergraph theory is
finding the maximum number of edges in a hypergraph
that does not contain a specific forbidden structure.
We consider as a forbidden structure a fixed number of members
that have empty common intersection
as well as small union.
We obtain a sharp upper bound on the size of uniform hypergraphs
that do not contain this structure,
when the number of vertices is sufficiently large.
Our result is strong enough to imply the same sharp upper bound
for several other interesting forbidden structures
such as the so-called strong simplices and clusters.
The {\em -dimensional hypercube}, ,
is the graph whose vertex set is and
whose edge set consists of the vertex pairs
differing in exactly one coordinate.
The generalized Tur\'an problem asks for the maximum number
of edges in a subgraph of a graph that does not contain
a forbidden subgraph .
We consider the Tur\'an problem where is and
is a cycle of length with .
Confirming a conjecture of Erd{\H o}s (1984),
we show that the ratio of the size of such a subgraph of
over the number of edges of is ,
i.e. in the limit this ratio approaches 0
as approaches infinity
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