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Problems Column 219

Discussiones Mathematicae Graph Theory 25 (2005 ) 219

COLORING OF G

2

\ G, FOR EUCLIDESIAN GRAPH G Konstanty Junosza-Szaniawski

Warsaw University of Technology Pl. Politechniki 1, 00–661 Warsaw, Poland

e-mail: k.szaniawski@mini.pw.edu.pl

The problem appeared in telecommunication.

Graph G = (V (G), E(G)) is called Euclidean if and only if V (G) is a finite subset of R

2

and {x, y} ∈ E(G) if and only if dist(x, y) ≤ d, where d ∈ R is fixed. Let S(G) = G

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\ G e.g. The vertex set of S(G) is V (G) and there is an edge {x, y} in E(S(G)) if and only if {x, y} / ∈ E(G) and x, y have a common neighbor in G. We consider vertex coloring of the graphs S(G), where G are Euclidean.

Problem 1. Is there a polynomial algorithm, which gives the chromatic number of S(G) for Euclidean graph G.

The problem appeared in telecommunication. In practical applications stan- dard approximate algorithms are used, but they do not use the geometric properties of S(G) and they seem not to be the most effective.

For geometric reasons χ(S(G)) ≤ 12, where G is Euclidean, but on other hand it is difficulty to find Euclidean graph G such that χ(S(G)) > 6.

Received 21 November 2003

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