## DISTANCE COLORING OF THE HEXAGONAL LATTICE

 Peter Jacko Universidad Carlos III de Madrid Department of Business Administration Calle Madrid 126, 289 03 Getafe (Madrid), Spain e-mail: peter.jacko@uc3m.es Stanislav Jendrol' P.J. Safárik University Institute of Mathematics Jesenná 5, 041 54 Košice, Slovakia e-mail: jendrol@Košice.upjs.sk

## Abstract

Motivated by the frequency assignment problem we study the d-distant coloring of the vertices of an infinite plane hexagonal lattice H. Let d be a positive integer. A d-distant coloring of the lattice H is a coloring of the vertices of H such that each pair of vertices distance at most d apart have different colors. The d-distant chromatic number of H, denoted χd(H), is the minimum number of colors needed for a d-distant coloring of H. We give the exact value of χd(H) for any d odd and estimations for any d even.

Keywords: distance coloring, distant chromatic number, hexagonal lattice of the plane, hexagonal tiling, hexagonal grid, radio channel frequency assignment.

2000 Mathematics Subject Classification: 05C15, 05C12.

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