12 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
Saturation in the Hypercube and Bootstrap Percolation
Let denote the hypercube of dimension . Given , a spanning
subgraph of is said to be -saturated if it does not
contain as a subgraph but adding any edge of
creates a copy of in . Answering a question of Johnson and Pinto, we
show that for every fixed the minimum number of edges in a
-saturated graph is .
We also study weak saturation, which is a form of bootstrap percolation. A
spanning subgraph of is said to be weakly -saturated if the
edges of can be added to one at a time so that each
added edge creates a new copy of . Answering another question of Johnson
and Pinto, we determine the minimum number of edges in a weakly
-saturated graph for all . More generally, we
determine the minimum number of edges in a subgraph of the -dimensional grid
which is weakly saturated with respect to `axis aligned' copies of a
smaller grid . We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and
Computin
Vertex Turán problems for the oriented hypercube
In this short note we consider the oriented vertex Turán problem in the hypercube: for a fixed oriented graph F→, determine the maximum size exv(F→,Qn−→) of a subset U of the vertices of the oriented hypercube Qn−→ such that the induced subgraph Qn−→[U] does not contain any copy of F→. We obtain the exact value of exv(Pk−→,Qn−→) for the directed path Pk−→, the exact value of exv(V2−→,Qn−→) for the directed cherry V2−→ and the asymptotic value of exv(T→,Qn−→) for any directed tree T→
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
Supersaturation of C4: From Zarankiewicz towards Erdős–Simonovits–Sidorenko
For a positive integer n, a graph F and a bipartite graph G subset of K-n,K-n let F(n + n, G) denote the number of copies of F in G, and let F(n + n, m) denote the minimum number of copies of F in all graphs G subset of K-n,K-n with m edges. The study of such a function is the subject of theorems of supersaturated graphs and closely related to the Sidorenko-Erdos-Simonovits conjecture as well. In the present paper we investigate the case when F = K-2.t and in particular the quadrilateral graph case. For F = C-4, we obtain exact results if m and the corresponding Zarankiewicz number differ by at most n, by a finite geometric construction of almost difference sets. F = K-2.t if m and the corresponding Zarankiewicz number differ by c . n root n we prove asymptotically sharp results based on a finite field construction. We also study stability questions and point out the connections to covering and packing block designs. (C) 2018 Elsevier Ltd. All rights reserved
Pairing strategies for the Maker-Breaker game on the hypercube with subcubes as winning sets
We consider the Maker-Breaker positional game on the vertices of the
-dimensional hypercube with -dimensional subcubes as winning
sets. We describe a pairing strategy which allows Breaker to win if is a
power of 4 and . Our results also imply that for all
there is a Breaker's win pairing strategy if .Comment: 24 page