5 citations · 10 across the 11 of their papers we have counts for
11 papers
The chromatic number of is at least 25
Jaan Parts
The lower bound for the chromatic number of the Euclidean space of dimension 8 is increased to 25.
More certainty in coloring the plane with a forbidden distance interval
Jaan Parts
In the mysterious and colorful world of chromatic numbers, where there are a lot of unknown, there is an amazing thing. It turns out that for some intervals of forbidden distances…
On lower bounds of the order of -chromatic unit distance graphs
Aubrey D. N. J. de Grey, Jaan Parts
Here we give refined numerical values for the minimum number of vertices of -chromatic unit distance graphs in the Euclidean plane.
Tiling the plane with hexagons: improved separations for -colourings
Aubrey D. N. J. de Grey, Jaan Parts
It has been common knowledge since 1950 that seven colours can be assigned to tiles of an infinite honeycomb with cells of unit diameter such that no two tiles of the same colour a…
On the plane and its coloring
Jaan Parts
We provide a human-verifiable proof that, in a certain sense, the chromatic number of the plane is exactly 7.
A 6-chromatic odd-distance graph in the plane
Jaan Parts
Two vertices of an odd-distance graph are connected by an edge if and only if their Euclidean distance is an odd integer. We construct a 6-chromatic odd-distance graph in the plane…
On upper bounds for the multi-fold chromatic numbers of the plane
Jaan Parts
The multi-fold chromatic number of the plane is the smallest number of colors , sufficient to color each point of the Euclidean plane in exactly colors, so that for an…
What percent of the plane can be properly 5- and 6-colored?
Jaan Parts
We present a tiling of more than 99.985698% of the Euclidean plane with six colors, reducing the previous record for uncovered fraction of the plane by about 12.8%. We also present…
The chromatic number of the plane is at least 5 -- a human-verifiable proof
Jaan Parts
We present a new proof of the known fact that the chromatic number of the plane is at least 5. The main difference of this proof is that it can be verified manually without the hel…
Graph minimization, focusing on the example of 5-chromatic unit-distance graphs in the plane
Jaan Parts
We introduce a new graph minimization method, in which it is required to preserve some graph property and there is an effective procedure for checking this property. We applied thi…
A small 6-chromatic two-distance graph in the plane
Jaan Parts
We give a new, simple proof for the lower bound of the chromatic number of the Euclidean plane with two forbidden distances, based on a graph with only 16 vertices.