# Serious applications of Colouring proofs

Are there any research-level applications of proofs by colouring?

This is the kind of proof you use to show that you can't cover a mutilated chessboard with 31 dominoes. Afaik, this technique chiefly finds a market in IMO preparation classes; see Arthur Engel's book or this handout .

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I don't think you can formally distinguish "proofs by coloring" from "parity arguments" or more generally arguments that use some mapping into a finite domain, say be reducing mod m. Coloring is just a visually appealing way of looking at such an argument. Needless to say, proofs by parity arguments or other mappings into a finite set are very common; this is a fundamental tool in combinatorics and number theory.

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Sperner's lemma has several applications, including the effective computation of fixed points and a constructive proof of the Brouwer fixed point theorem.

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you can use this syntax for links: [link](url), as in Conway's Soldiers. –  lhf May 17 '11 at 10:47