Bartosz Maria Walczak

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Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the ...

In the 1970s Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer k we construct a triangle-free fam ...

A family of sets in the plane is simple if the intersection of any subfamily is arc-connected, and it is pierced by a line L if the intersection of any member with L is a nonempty segment. It is proved that the intersection graphs of simple families of com ...

Suppose k is a positive integer and 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(n(2)) with the property that any ...

The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs defined by geometric representations and competitivity analysis of on-line algorithms. This connection became apparent after the recent construction ...

We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that Delta - 1 edge slopes suffic ...

For positive integers w and k, two vectors A and B from Z(w) are called k-crossing if there are two coordinates i and j such that A[i] - B[i] >= k and B[j] - A[j] >= k. What is the maximum size of a family of pairwise 1-crossing and pairwise non-k-crossing ...

Bartosz Maria Walczak

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Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Triangle-free intersection graphs of line segments with large chromatic number

Coloring Intersection Graphs of Arc-Connected Sets in the Plane

Decomposition of Multiple Packings with Subquadratic Union Complexity

Coloring Relatives of Interval Overlap Graphs via On-line Games

Outerplanar graph drawings with few slopes

An extremal problem on crossing vectors