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)
Cesium-129 is concentrated in the myocardium after intravenous administration permitting myocardial imaging. The dosage used was 2-2.5 mCi in dogs and 3-4 mCi in patients. Four or more views with 200,000 counts per view were obtained 30 to 90 minutes after administration. Control images were obtained in 30 dogs. In two dogs anatomic landmarks were obtained(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)
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 than .(More)
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. A coloring of a graph G is called (C1, C2)-relaxed if every monochromatic component induced by vertices of the first (second) color is of order at most C1 (C2, resp.). We prove that the(More)