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Decomposition of Multiple Packings with Subquadratic Union Complexity

Published online by Cambridge University Press:  04 January 2016

JÁNOS PACH  and
BARTOSZ WALCZAK
JÁNOS PACH
Affiliation:
École Polytechnique Fédérale de Lausanne, Switzerland and Alfred Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary (e-mail: pach@cims.nyu.edu)
BARTOSZ WALCZAK
Affiliation:
Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland (e-mail: walczak@tcs.uj.edu.pl)
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Abstract

Suppose k is a positive integer and ${\cal X}$ is a k-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most k sets. Suppose there is a function f(n) = o(n2) with the property that any n members of ${\cal X}$ determine at most f(n) holes, which means that the complement of their union has at most f(n) bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that ${\cal X}$ can be decomposed into at most p (1-fold) packings, where p is a constant depending only on k and f.

Keywords

Primary 52C15Secondary 05B40
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 1: Oberwolfach Special Issue Part 2 , January 2016 , pp. 145 - 153
Copyright
Copyright © Cambridge University Press 2015 

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