52 research outputs found
Extensions of Extremal Graph Theory to Grids
We consider extensions of Turán\u27s original theorem of 1941 to planar grids. For a complete kxm array of vertices, we establish in Proposition 4.3 an exact formula for the maximal number of edges possible without any square regions. We establish with Theorem 4.12 an upper bound and with Theorem 4.15 an asymptotic lower bound for the maximal number of edges on a general grid graph with n vertices and no rectangles
On Some Applications of Graph Theory, I
In a series of papers, of which the present one is Part 1, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results. (c) 1972 Published by Elsevier B.V
Some Ramsey- and anti-Ramsey-type results in combinatorial number theory and geometry
A szerző nem járult hozzá nyilatkozatában a dolgozat nyilvánosságra hozásához
On Minrank and Forbidden Subgraphs
The minrank over a field of a graph on the vertex set
is the minimum possible rank of a matrix such that for every , and
for every distinct non-adjacent vertices and in . For an
integer , a graph , and a field , let
denote the maximum possible minrank over of an -vertex graph
whose complement contains no copy of . In this paper we study this quantity
for various graphs and fields . For finite fields, we prove by
a probabilistic argument a general lower bound on , which
yields a nearly tight bound of for the triangle
. For the real field, we prove by an explicit construction that for
every non-bipartite graph , for some
. As a by-product of this construction, we disprove a
conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by
questions in information theory, circuit complexity, and geometry.Comment: 15 page
On the Ramsey-Tur\'an density of triangles
One of the oldest results in modern graph theory, due to Mantel, asserts that
every triangle-free graphs on vertices has at most
edges. About half a century later Andr\'asfai studied dense triangle-free
graphs and proved that the largest triangle-free graphs on vertices without
independent sets of size , where , are blow-ups
of the pentagon. More than 50 further years have elapsed since Andr\'asfai's
work. In this article we make the next step towards understanding the structure
of dense triangle-free graphs without large independent sets.
Notably, we determine the maximum size of triangle-free graphs~ on
vertices with and state a conjecture on the structure of
the densest triangle-free graphs with . We remark that the
case behaves differently, but due to the work of Brandt
this situation is fairly well understood.Comment: Revised according to referee report
On Grids in Point-Line Arrangements in the Plane
The famous Szemer\'{e}di-Trotter theorem states that any arrangement of
points and lines in the plane determines incidences, and this
bound is tight. In this paper, we prove the following Tur\'an-type result for
point-line incidence. Let and be two sets of
lines in the plane and let be the set of intersection points
between and . We say that forms a \emph{natural grid} if , and
does not contain the intersection point of some two lines in
for For fixed , we show that any arrangement
of points and lines in the plane that does not contain a natural
grid determines incidences, where
. We also provide a construction of points
and lines in the plane that does not contain a natural grid
and determines at least incidences.Comment: 13 pages, 5 figure
On triangle-free graphs maximizing embeddings of bipartite graphs
In 1991 Gy\H ori, Pach, and Simonovits proved that for any bipartite graph
containing a matching avoiding at most 1 vertex, the maximum number of
copies of in any large enough triangle-free graph is achieved in a balanced
complete bipartite graph. In this paper we improve their result by showing that
if is a bipartite graph containing a matching of size and at most
unmatched vertices, then the maximum number of copies
of in any large enough triangle-free graph is achieved in a complete
bipartite graph. We also prove that such a statement cannot hold if the number
of unmatched vertices is
On some applications of graph theory, I.
In a series of papers, of which the present one is Part I, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results. © 1972
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