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For a graph whose vertex set is a finite set of points in R d , consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect. Using the idea of halving lines, we show that (i) for any finite set of points in the plane, there exists a Hamiltonian cycle that is a Tverberg graph; (ii) for any n red and n blue points in the plane, there exists a perfect red-blue matching that is a Tverberg graph. Using the idea of infinite descent, we prove that (iii) for any even set of points in R d , there exists a perfect matching that is an open Tverberg graph; (iv ) for any n red and n blue points in R d , there exists a perfect red-blue matching that is a Tverberg graph.

The Len~nergo system in Leningrad has six hydroelectric plants with a total capacity of 700 MW, which represents over 1.5% of the overaU capacity of this power system. The hydroelectric plants operate basically during the peaking portion of the load graph of the system, with one or two interruptions during the day. Under these operating conditions, there are regimen and technical output losses, caused by a decrease in the head and the efficiency as a resuk of regulation and of increased head losses in the trashracks. On the whole, in this power system the total energy losses at the hydroelectric plants amount to 6o7~ and their absolute value, even for the low-water years 1972 and 1973, exceeds 120 million kwh. The distribution of the additional energy losses at the different sources is as follows (according to data for 1972): in the trashracks, 1.3%; during daily regulation because of head drops, 2.2%; during regulation because of decrease in the efficiency, 0.25%; other losses (idle overflows, locking, seepage), 1.1%.Taking into account the increase in the daily regulation losses, caused by increase in the participation of the hydroelectric plants in regimen regulation the operating personnel of these plants and the Power Board focused considerable attention on reducing the technical losses, especially the head losses in the trashracks.Under the regimen exigencies of the power system, where basically a power effect and maximum participation in the covering of peak loads are needed from the hydroelectric plants, the head losses in the trashracks and the penstocks lead to an appreciable power reduction precisely during peaking operation, when maximum loads are placed on the hydroelectric plants and there are maximum discharges through the turbines. The high flow velocity and the rapid increase in the plant loads contribute to further intense obstruction of the trashracks. It follows that the problem of reducing the above-mentioned head losses is associated with the improvement of the operating flexibility of the hydroelectric plants and the increase in their participation in the regimen regulation.From the viewpoint of construction of the trashracks, the streams on which the I.en~nergo hydroelectric plants are located exhibit extremely dissimilar characteristics. For example, during the summer period, along the upper reaches of the Svir' River in the Ivinsk overflow of the reservoir for the Upper Svir' hydroelectric plant, large peat masses float and become peat islands, which are very harmful to hydroelectric plant operation. The Volkhov River, which flows through habitable and populated areas, carries during the flood period bushes, grass, construction debris, and different objects which obstruct the trashracks and bring up to 1.0 m and over the head loss through them. During the summer a large volume of grass is conveyed from the shallow Narva reservoir to the penstocks of the Narva hydroelectric plant. Under more favorable conditions, from the viewpoint of trashrack obstruction , are the Svetogorsk and Lesogor...

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