271 research outputs found
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids
The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest
integer k such that for every colouring of the vertices of H with exactly k
colours, there is a hyperedge of H all of whose vertices have different
colours. We denote by nu(H) the number of vertices of H and by tau(H) the size
of the smallest set containing at least two vertices of each hyperedge of H.
For a complete geometric graph G with n > 2 vertices let H = H(G) be the
hypergraph whose vertices are the edges of G and whose hyperedges are the edge
sets of plane spanning trees of G. We prove that if G has at most one interior
vertex, then hc(H) = nu(H) - tau(H) + 2. We also show that hc(H) = nu(H) -
tau(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given
by the ground set and the bases of a matroid, respectively
Ramsey Theory Using Matroid Minors
This thesis considers a Ramsey Theory question for graphs and regular matroids. Specifically, how many elements N are required in a 3-connected graphic or regular matroid to force the existence of certain specified minors in that matroid? This question cannot be answered for an arbitrary collection of specified minors. However, there are results from the literature for which the number N exists for certain collections of minors. We first encode totally unimodular matrix representations of certain matroids. We use the computer program MACEK to investigate this question for certain classes of specified minors
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
Some remarks on off-diagonal Ramsey numbers for vector spaces over
For every positive integer , we show that there must exist an absolute
constant such that the following holds: for any integer
and any red-blue coloring of the one-dimensional subspaces of
, there must exist either a -dimensional subspace for
which all of its one-dimensional subspaces get colored red or a -dimensional
subspace for which all of its one-dimensional subspaces get colored blue. This
answers recent questions of Nelson and Nomoto, and confirms that for any even
plane binary matroid , the class of -free, claw-free binary matroids is
polynomially -bounded.
Our argument will proceed via a reduction to a well-studied additive
combinatorics problem, originally posed by Green: given a set with density , what is the largest
subspace that we can find in ? Our main contribution to the story is a new
result for this problem in the regime where is large with respect to
, which utilizes ideas from the recent breakthrough paper of Kelley and Meka
on sets of integers without three-term arithmetic progressions
A note on order-type homogeneous point sets
Let OT_d(n) be the smallest integer N such that every N-element point
sequence in R^d in general position contains an order-type homogeneous subset
of size n, where a set is order-type homogeneous if all (d+1)-tuples from this
set have the same orientation. It is known that a point sequence in R^d that is
order-type homogeneous forms the vertex set of a convex polytope that is
combinatorially equivalent to a cyclic polytope in R^d. Two famous theorems of
Erdos and Szekeres from 1935 imply that OT_1(n) = Theta(n^2) and OT_2(n) =
2^(Theta(n)). For d \geq 3, we give new bounds for OT_d(n). In particular:
1. We show that OT_3(n) = 2^(2^(Theta(n))), answering a question of
Eli\'a\v{s} and Matou\v{s}ek.
2. For d \geq 4, we show that OT_d(n) is bounded above by an exponential
tower of height d with O(n) in the topmost exponent
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