Graph colouring — the series
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Four 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.
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Five 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.
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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.
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Seven 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.
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The order decides the colours
The simplest way to colour a graph is to take the vertices one at a time and give each the first colour its neighbours are not already using. It never needs more than one colour beyond the largest degree — and on a graph that needs only two colours it can be made to use as many as there are vertices on a side, depending on nothing but the order it is handed.
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How many colours the plane needs
Colour every point of the plane so that no two points exactly one unit apart share a colour. Seven colours are enough, by a tiling with hexagons, and four are necessary, by a graph of seven points. For sixty-eight years nothing better was known on either side; then in 2018 a graph of 1,581 points showed four is not enough — and the answer is still somewhere from five to seven.
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No triangle, and four colours
A graph that contains four mutually joined points needs four colours. The converse looks obvious — a graph needing four colours should contain something like that — and it is false. Grötzsch's graph has eleven points, no three of them mutually joined, and needs four colours. Mycielski found a step that adds one colour to any graph without creating a triangle, so graphs with no triangle can need any number of colours, and their need is spread through the whole graph, with no part to point at.