Discussiones Mathematicae Graph Theory 25(1-2) (2005) 151-166
doi: 10.7151/dmgt.1269

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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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Received 25 November 2003
Revised 22 February 2005