The Hadwiger-Nelson Problem

The computers like copying the graph, and the center of it has four colours. They pile it until 32291925 vertices. There are five colours. This is called the Hadwiger-Nelson problem. You need to choose the minimum value.

This is the graph H, which is the center of the expansion. There are 4 colours. The center is yellow, but the distance of 1 must be the different colours.

This is the graph J, which contains 31 vertices and 13 copies of H.

Then you change the color.

This is different from J. The center is H, and there are 6 copies of H. The distance from the center is √3.

Moreover, black vertices are included, so they are different hexagons from H. There is no same one.

There are 7 hexagons. This chromatic number is 7.

This is the graph K, which has 61 vertices. This is 2 copies of J.

This is the graph L, which has 121 vertices. This is 2 copies of K.

This is M which has 20425 vertices.

This is G which has 32291925 vertices. This is very complicated and highly dense. Therefore, it is hard to find the four coloured core in recent computing.

You remember that this is the center of G, so there must be 5 colours. The distance of black vertices is 1, but they can assume any colour so long as no connected vertices are the same colour.