71 research outputs found
Coloring geometric hyper-graph defined by an arrangement of half-planes
We prove that any finite set of half-planes can be colored by two colors so
that every point of the plane, which belongs to at least three half-planes in
the set, is covered by half-planes of both colors. This settles a problem of
Keszegh
Homometric sets in trees
Let denote a simple graph with the vertex set and the edge
set . The profile of a vertex set denotes the multiset of
pairwise distances between the vertices of . Two disjoint subsets of
are \emph{homometric}, if their profiles are the same. If is a tree on
vertices we prove that its vertex sets contains a pair of disjoint homometric
subsets of size at least . Previously it was known that such a
pair of size at least roughly exists. We get a better result in case
of haircomb trees, in which we are able to find a pair of disjoint homometric
sets of size at least for a constant
The -genus of Kuratowski minors
A drawing of a graph on a surface is independently even if every pair of
nonadjacent edges in the drawing crosses an even number of times. The
-genus of a graph is the minimum such that has an
independently even drawing on the orientable surface of genus . An
unpublished result by Robertson and Seymour implies that for every , every
graph of sufficiently large genus contains as a minor a projective
grid or one of the following so-called -Kuratowski graphs: , or
copies of or sharing at most common vertices. We show that
the -genus of graphs in these families is unbounded in ; in
fact, equal to their genus. Together, this implies that the genus of a graph is
bounded from above by a function of its -genus, solving a problem
posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of
the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous
result for Euler genus and Euler -genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte
A computational approach to Conway's thrackle conjecture
A drawing of a graph in the plane is called a thrackle if every pair of edges
meets precisely once, either at a common vertex or at a proper crossing. Let
t(n) denote the maximum number of edges that a thrackle of n vertices can have.
According to a 40 years old conjecture of Conway, t(n)=n for every n>2. For any
eps>0, we give an algorithm terminating in e^{O((1/eps^2)ln(1/eps))} steps to
decide whether t(n)2. Using this approach, we improve the
best known upper bound, t(n)<=3/2(n-1), due to Cairns and Nikolayevsky, to
167/117n<1.428n.Comment: 16 pages, 7 figure
Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4
We find a graph of genus and its drawing on the orientable surface of
genus with every pair of independent edges crossing an even number of
times. This shows that the strong Hanani-Tutte theorem cannot be extended to
the orientable surface of genus . As a base step in the construction we use
a counterexample to an extension of the unified Hanani-Tutte theorem on the
torus.Comment: 12 pages, 4 figures; minor revision, new section on open problem
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