A wheel of 5 rim regions needs 4 colours
map-colour is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
Counting the colourings, by deleting and contracting
One swap frees a colour
Seven regions on a doughnut, each touching all six others
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- region 0 touches region 1 ×21
- deleting and contracting account for the count at 0 colours ×6
- a colour has been freed for the middle ×1
- after: no edge joins two regions of one colour ×1
- an odd rim needs four and an even rim three ×1
- before: no edge joins two regions of one colour ×1
- between 3 and 6 rows are drawn ×1
- colours are counted up to between 3 and 7 ×1
- every one of the seven regions touches every other ×1
- four colours are enough ×1
- no graph with an edge can be coloured with no colours ×1
- no two neighbouring regions share a colour ×1
- nor with one, when it has an edge ×1
- the case is the one where the chain reaches or the one where it does not ×1
- the chain reaches the far neighbour exactly in the blocked case ×1
- the chain that is swapped does not reach the other neighbour of its pair ×1
- the chromatic number is where the count first becomes positive ×1
- the construction shown is the seven-region one ×1
- the five neighbours use five different colours, which is the only hard case ×1
- the graph is one the family draws ×1
- the map can be coloured with at most five ×1
- the map has between 4 and 24 regions ×1
- the vertical step is 2, 3 or 4 ×1
- the view is one the family draws ×1
- the wheel has between 3 and 23 rim regions ×1
- with the middle coloured: no edge joins two regions of one colour ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
Counting the colourings
Asking whether a graph can be coloured with four colours gives a yes or a no. Asking how many ways there are gives a polynomial — and the polynomial answers the first question, and several others nobody asked.
DiscreteFive colours, and a chain that can be followed
The four-colour theorem cannot be checked by a person. The five-colour theorem can, in a page, and the argument that does it is the one Kempe thought had settled four — with the exact step where it fails visible in the picture.
DiscreteFour colours, and a proof nobody can read
Every map on a plane can be coloured with four colours so that no two neighbours match. The statement is understandable by a child, it resisted a century of attempts, and the proof that settled it cannot be checked by a human being.
DiscreteSeven regions on a doughnut
A map on a torus can need seven colours, and the proof is a picture — seven regions, each sharing a border with all six others. The plane needed a computer and eighty-six years; the harder surface was settled in 1890 by drawing something.
DiscreteTwo graphs that will not lie flat
Five points, every pair joined: no matter how the points are placed or how the lines are drawn, two of the lines cross. The proof is not about drawing at all — it counts edges against faces and finds one edge too many.