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Patch Colorings and Rigid Colorings of the Rational n-Space

✍ Scribed by Grzegorz Kubicki; Jenő Lehel; Michał Morayne

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
108 KB
Volume
84
Category
Article
ISSN
0095-8956

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✦ Synopsis

We investigate k-colorings of the rational n-space, Q n , such that any two points at distance one get distinct colors. Two types of colorings are considered: patch colorings where the colors occupy open sets with parts of their boundary, and rigid colorings which uniquely extend from any open subset of Q n . We prove that the existence of a patch k-coloring of Q n implies the existence of a k-coloring of R n . We show that every 2-coloring of Q 2 or Q 3 , and every 4-coloring of Q 4 is rigid. We also construct a rigid 3-coloring of Q 2 .

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