TY - JOUR

T1 - Intersection reverse sequences and geometric applications

AU - Marcus, Adam

AU - Tardos, Gábor

N1 - Funding Information:
E-mail addresses: adam@math.gatech.edu (A. Marcus), tardos@renyi.hu (G. Tardos). 1 On leave from the Department of Mathematics (ACO), Georgia Institute of Technology, Atlanta, GA 30332-0160. This research was made possible due to funding by the Fulbright Program in Hungary. 2Partially supported by the Hungarian National Research Fund grants OTKA T048826, OTKA T046234 and OTKA T037846.

PY - 2006/5

Y1 - 2006/5

N2 - Pinchasi and Radoičić [On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-crossing C4 has O(n3/2 log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n3/2 log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.

AB - Pinchasi and Radoičić [On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-crossing C4 has O(n3/2 log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n3/2 log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.

KW - Cyclic order

KW - Extremal combinatorics

KW - Pseudocircles

KW - Topological graphs

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U2 - 10.1016/j.jcta.2005.07.002

DO - 10.1016/j.jcta.2005.07.002

M3 - Article

AN - SCOPUS:33645550027

SN - 0097-3165

VL - 113

SP - 675

EP - 691

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 4

ER -