A Tura´n-type theorem on chords of a convex polygon
Journal of Combinatorial Theory Series B
Approximation schemes for covering and packing problems in image processing and VLSI
Journal of the ACM (JACM)
Label placement by maximum independent set in rectangles
WADS '97 Selected papers presented at the international workshop on Algorithms and data structure
Journal of Combinatorial Theory Series A
Excluded permutation matrices and the Stanley-Wilf conjecture
Journal of Combinatorial Theory Series A
Crossing number, pair-crossing number, and expansion
Journal of Combinatorial Theory Series B
Topological Graphs with No Large Grids
Graphs and Combinatorics
Crossing Stars in Topological Graphs
SIAM Journal on Discrete Mathematics
Coloring kk-free intersection graphs of geometric objects in the plane
Proceedings of the twenty-fourth annual symposium on Computational geometry
A bipartite strengthening of the crossing lemma
GD'07 Proceedings of the 15th international conference on Graph drawing
Disjoint edges in topological graphs
IJCCGGT'03 Proceedings of the 2003 Indonesia-Japan joint conference on Combinatorial Geometry and Graph Theory
GD'11 Proceedings of the 19th international conference on Graph Drawing
String graphs and incomparability graphs
Proceedings of the twenty-eighth annual symposium on Computational geometry
Hi-index | 0.00 |
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 log2 n factors, respectively. We also settle the conjectures for some special cases.