Robert Berke

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We show that the vertices of any plane graph in which every face is of size at least <i>g</i> can be colored by (3g &#192;&#253; 5)=4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g+1)=4 colors. We further show that the problem of determining(More)
  • Leonard Yves, R Ust, M Sc, Eth In, Eth Bernd Gärtner, Zurich +40 others
  • 2007
The goal of this thesis is to give a better understanding of the linear complementarity problem with a P-matrix (PLCP). Finding a polynomial time algorithm for the PLCP is a longstanding open problem. Such an algorithm would settle the complexity status of many problems reducing to the PLCP. Most of the papers dealing with the PLCP look at it from an(More)
We show that any graph of maximum degree at most ¢ has a two-coloring, such that one color-class is an independent set while the other color induces monochromatic components of order at most £ ¥ ¤ § ¦. On the other hand for any constant¨we exhibit a ©-regular graph, such that the deletion of any independent set leaves at least one component of order greater(More)
A coloring is proper if each color class induces connected components of order one (where the order of a graph is its number of vertices). Here we study relaxations of proper two-colorings, such that the order of the induced monochromatic components in one (or both) of the color classes is bounded by a constant. In a (C1, C2)-relaxed coloring of a graph G(More)
We study relaxations of proper two-colorings, such that the order of the induced monochro-matic components in one (or both) of the color classes is bounded by a constant. A coloring of a graph G is called (C 1 , C 2)-relaxed if every monochromatic component induced by vertices of the first (second) color is of order at most C 1 (C 2 , resp.). We prove that(More)
2009 to my parents Acknowledgments I am deeply grateful to my supervisor Emo Welzl for his motivating and encouraging support during my time as Ph.D. student. He introduced me to the beautiful field of geometric graph theory, and with his inspiring advice he made it an unforgettable experience to learn from him. It was an honor being part of his research(More)
OBJECTIVE To compare the investigator assessment of patient risk for prescription opioid misuse, abuse, and diversion with patient self-reports of these activities in a population with chronic pain. METHODS As a secondary objective of an open-label, multicenter, primary care-based clinical study to evaluate the success of converting opioid-experienced(More)
OBJECTIVE To evaluate the conversion of opioid-experienced patients with chronic moderate-to-severe pain to extended-release morphine sulfate with sequestered naltrexone hydrochloride (MSN) using a standardized conversion guide. METHODS This open-label, single-arm study was conducted in 157 primary care centers in the United States. A total of 684(More)
Transversals in r-partite graphs with various properties are known to have many applications in graph theory and theoretical computer science. We investigate f-bounded transversals (or f-BT), that is, transversals whose connected components have order at most f. In some sense we search for the the sparsest f-BT-free graphs. We obtain estimates on the(More)