Kolja B. Knauer

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We investigate edge-intersection graphs of paths in the plane grid regarding a parameter called the bend-number. The bend-number is related to the interval-number and the track-number of a graph. We provide new upper and lower bounds of the bend-number of any given simple graph in terms of the coloring number, edge clique covers and the maximum degree. We(More)
We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that d 2 e edge slopes suffice for outerplanar drawings of outerplanar graphs with maximum degree ∆ > 3. This matches the obvious lower bound. We also(More)
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 ∆ − 1 edge slopes suffice for every outerplanar graph with maximum degree ∆ > 4. This improves on the previous bound of O(∆), which(More)
We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R can be colored with k colors so that every translate of the negative octant containing at least k points contains at least one of each color. The(More)
We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive(More)
An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, L , Land L . A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an L, an L or L , a k-bend(More)
The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of(More)
Two players want to eat a sliced pizza by alternately picking its pieces. The pieces may be of various sizes. After the first piece is eaten every subsequently picked piece must be adjacent to some previously eaten. We provide a strategy for the starting player to eat 4 9 of the total size of the pizza. This is best possible and settles a conjecture of(More)