13 research outputs found
Extremal numbers for cycles in a hypercube
Let be the largest number of edges in a subgraph of a
hypercube such that there is no subgraph of isomorphic to . We
show that for any integer , Comment: New reference [18] for a better bound by I.Tomon is adde
A class of graphs of zero Tur\'an density in a hypercube
A graph is cubical if it is a subgraph of a hypercube. For a cubical graph
and a hypercube , is the largest number of edges in an
-free subgraph of . If is at least a positive proportion
of the number of edges in , is said to have a positive Tur\'an density
in a hypercube or simply a positive Tur\'an density; otherwise it has a zero
Tur\'an density. Determining and even identifying whether has
a positive or a zero Tur\'an density remains a widely open question for general
. By relating extremal numbers in a hypercube and certain corresponding
hypergraphs, Conlon found a large class of cubical graphs, ones having
so-called partite representation, that have a zero Tur\'an density. He raised a
question whether this gives a characterisation, i.e., whether a cubical graph
has zero Tur\'an density if and only if it has partite representation. Here, we
show that, as suspected by Conlon, this is not the case. We give an example of
a class of cubical graphs which have no partite representation, but on the
other hand, have a zero Tur\'an density. In addition, we show that any graph
whose every block has partite representation has a zero Tur\'an density in a
hypercube
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
On graphs embeddable in a layer of a hypercube and their extremal numbers
A graph is cubical if it is a subgraph of a hypercube. For a cubical graph
and a hypercube , is the largest number of edges in an
-free subgraph of . If is equal to a positive proportion
of the number of edges in , is said to have positive Tur\'an density
in a hypercube; otherwise it has zero Tur\'an density. Determining
and even identifying whether has positive or zero Tur\'an density remains a
widely open question for general .
In this paper we focus on layered graphs, i.e., graphs that are contained in
an edge-layer of some hypercube. Graphs that are not layered have positive
Tur\'an density because one can form an -free subgraph of consisting
of edges of every other layer. For example, a -cycle is not layered and has
positive Tur\'an density.
However, in general it is not obvious what properties layered graphs have. We
give a characterisation of layered graphs in terms of edge-colorings and show
that any -vertex layered graphs has at most
edges. We show that most non-trivial subdivisions have zero Tur\'an density,
extending known results on zero Tur\'an density of even cycles of length at
least and of length . However, we prove that there are cubical graphs
of girth that are not layered and thus having positive Tur\'an density. The
cycle of length remains the only cycle for which it is not known whether
its Tur\'an density is positive or not. We prove that , for a constant , showing that the extremal number
for a -cycle behaves differently from any other cycle of zero Tur\'an
density
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→