On multiple coverings of the infinite rectangular grid with balls of constant radius

  • Authors:

  • Maria A. Axenovich

  • Affiliations:

  • Department of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA and University of Illinois, Urbana-Champaign, Illinois

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2003

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Abstract

We consider the coverings of graphs with balls of constant radius satisfying special multiplicity condition. A (t, i, j)-cover of a graph G = (V, E) is a subset S of vertices, such that every element of S belongs to exactly i balls of radius t centered at elements of S and every element of V\S belongs to exactly j balls of radius t centered at elements of S. For the infinite rectangular grid, we show that in any (t, i, j)-cover, i and j differ by at most t + 2 except for one degenerate case. Furthermore, for i and j satisfying |i - j| 4 we show that all (t, i, j)-covers are the unions of the diagonals periodically located in the grid. Also, we give the description of all (1, i, j)-covers.