1 research outputs found
Chromatic properties of the Euclidean plane
Let be the unit distance graph in the plane. A well-known problem in
combinatorial geometry is that of determining the chromatic number of . It
is known that . The upper bound of 7 is obtained using
tilings of the plane. The present paper studies two problems where we seek
proper colourings of , adding restrictions inspired by tilings:
Let be the graph whose vertices are the points of , with an edge between two points if their distance lies in the interval
. We show that for small , , we have . This improves
the result of Exoo and Grytczuk et al. that for small
.
Suppose that is properly coloured, but so that two solidly coloured
regions meet along a straight line in some neighbourhood. Then at least 5
colours must be used.Comment: 11 pages, 6 figures. Reference to Grytczuk et al. added, March 22,
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