9,658 research outputs found
On the Number of Edges of Fan-Crossing Free Graphs
A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1
if there are no k+1 edges , such that have a
common endpoint and crosses all . We prove a tight bound of 4n-8 on
the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9
bound for a straight-edge drawing. For k > 2, we prove an upper bound of
3(k-1)(n-2) edges. We also discuss generalizations to monotone graph
properties
Applications of a new separator theorem for string graphs
An intersection graph of curves in the plane is called a string graph.
Matousek almost completely settled a conjecture of the authors by showing that
every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log
m). In the present note, this bound is combined with a result of the authors,
according to which every dense string graph contains a large complete balanced
bipartite graph. Three applications are given concerning string graphs G with n
vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic
number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph
of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided
Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds
for string graphs.Comment: 7 page
Disjoint edges in topological graphs and the tangled-thrackle conjecture
It is shown that for a constant , every simple topological
graph on vertices has edges if it has no two sets of edges such
that every edge in one set is disjoint from all edges of the other set (i.e.,
the complement of the intersection graph of the edges is -free). As an
application, we settle the \emph{tangled-thrackle} conjecture formulated by
Pach, Radoi\v{c}i\'c, and T\'oth: Every -vertex graph drawn in the plane
such that every pair of edges have precisely one point in common, where this
point is either a common endpoint, a crossing, or a point of tangency, has at
most edges
Edge Partitions of Optimal -plane and -plane Graphs
A topological graph is a graph drawn in the plane. A topological graph is
-plane, , if each edge is crossed at most times. We study the
problem of partitioning the edges of a -plane graph such that each partite
set forms a graph with a simpler structure. While this problem has been studied
for , we focus on optimal -plane and -plane graphs, which are
-plane and -plane graphs with maximum density. We prove the following
results. (i) It is not possible to partition the edges of a simple optimal
-plane graph into a -plane graph and a forest, while (ii) an edge
partition formed by a -plane graph and two plane forests always exists and
can be computed in linear time. (iii) We describe efficient algorithms to
partition the edges of a simple optimal -plane graph into a -plane graph
and a plane graph with maximum vertex degree , or with maximum vertex
degree if the optimal -plane graph is such that its crossing-free edges
form a graph with no separating triangles. (iv) We exhibit an infinite family
of simple optimal -plane graphs such that in any edge partition composed of
a -plane graph and a plane graph, the plane graph has maximum vertex degree
at least and the -plane graph has maximum vertex degree at least .
(v) We show that every optimal -plane graph whose crossing-free edges form a
biconnected graph can be decomposed, in linear time, into a -plane graph and
two plane forests
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