On some problems of covering the plane with circles

We study regular circle plane coverings in which the plane is split into regular polygons (tiles) and all the tiles are covered identically. The density of a regular covering can be calculated by dividing total area of circles covering a tile by the tile's area. We focus on regular coverings containing circles of four, five and six different radii. We prove optimality of several known coverings in their classes, find tight lower bounds for densities depending on radii of circles in a covering, and propose new coverings which are optimal in their classes under some additional constraints on radii. Ill. 14, bibliogr. 15.