TY - GEN

T1 - On grids in topological graphs

AU - Ackerman, Eyal

AU - Fox, Jacob

AU - Pach, János

AU - Suk, Andrew

PY - 2009

Y1 - 2009

N2 - A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges. We conjecture that this statement remains true (1) for topological graphs in which only k-grids consisting of 2k vertex-disjoint edges are forbidden, and (2) for graphs drawn by straight-line edges, with no k-element sets of edges such that every edge in the first set crosses every edge in the other set and each pair of edges within the same set is disjoint. These conjectures are shown to be true apart from log* n and log 2 n factors, respectively. We also settle the conjectures for some special cases.

AB - A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges. We conjecture that this statement remains true (1) for topological graphs in which only k-grids consisting of 2k vertex-disjoint edges are forbidden, and (2) for graphs drawn by straight-line edges, with no k-element sets of edges such that every edge in the first set crosses every edge in the other set and each pair of edges within the same set is disjoint. These conjectures are shown to be true apart from log* n and log 2 n factors, respectively. We also settle the conjectures for some special cases.

KW - Geometric graphs

KW - Topological graphs

KW - Turán-type problems

UR - http://www.scopus.com/inward/record.url?scp=70849133747&partnerID=8YFLogxK

U2 - 10.1145/1542362.1542430

DO - 10.1145/1542362.1542430

M3 - Conference contribution

AN - SCOPUS:70849133747

SN - 9781605585017

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 403

EP - 412

BT - Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09

T2 - 25th Annual Symposium on Computational Geometry, SCG'09

Y2 - 8 June 2009 through 10 June 2009

ER -