On lower bounds of the order of $k$-chromatic unit distance graphs
Here we give refined numerical values for the minimum number of vertices of $k$-chromatic unit distance graphs in the Euclidean plane.
math.CO↗
arXiv subjects
Aubrey D. N. J. de Grey
Publications and source records attributed to Aubrey D. N. J. de Grey.
Tiling the plane with hexagons: improved separations for $k$-colourings
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 are closer than $d(7)=\frac{\sqrt{7}}{2}$ apart. Various authors have described tilings using $k>7$ colours, giving corresponding values for $d(k)$, but it is generally unknown whether these are the largest possible for a given $k$. Here, for many $k$, we describe tilings with larger values of $d(k)$ than previously reported.
math.CO↗
The chromatic number of the plane is at least 5
We present a family of finite unit-distance graphs in the plane that are not 4-colourable, thereby improving the lower bound of the Hadwiger-Nelson problem. The smallest such graph that we have so far discovered has 1581 vertices.
math.CO↗