784 research outputs found
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
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
- …