‘Rainbows’ Are a Mathematician’s Best Friend | Mathematics - Quanta Magazine

Kevin Hartnett, senior writer at Quanta Magazine argues, “Rainbow colorings” recently led to a new proof. It’s not the first time they’ve come in handy.

Photo: Color-coding a Latin square and its graph can reveal a lot about them.

Recently, Quanta reported on the new solution to a problem called Ringel’s conjecture. Part of the proof involved using rainbow colorings, special color-coded ways of visualizing information. But the colorful technique has actually been helping mathematicians solve puzzles for a long time, and it figures in an even harder related problem that mathematicians are eyeing next.

Ringel’s conjecture is a problem in combinatorics where you connect dots (vertices) with lines (edges) to form graphs. It predicts that there’s a special relationship between a type of large graph with 2n + 1 vertices and a proportionally smaller type of graph with n + 1 vertices...

Eventually, mathematicians discovered that one way to investigate this is to turn the square into a graph. To do this, place three vertices on the left side of the page, representing the three columns. Then place three vertices on the right side of the page, representing the rows. Draw edges connecting each vertex on the right with each vertex on the left. Each edge, being the combination of a specific row and column, represents one of the nine boxes. For example, the edge between the top vertex on the right and the top vertex on the left corresponds to the box in the first row and the first column (the top left box in the Latin square)...

However, the methods that came to fruition in the Ringel proof seem likely to be applicable to graceful labeling — and mathematicians are eager to see just how far they can push them.
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Source: Quanta Magazine