Dense triangle-free graphs are four-colorable : A solution to the Erdős-Simonovits problem

In 1972, Erdős and Simonovits [9] asked whether a triangle-free graph with minimum degree greater than n/3, where n is the number of vertices, has chromatic number at most three. Hajnal provided examples of triangle-free graphs with arbitrarily large chromatic number and minimum degree greater than (1/3− ε)n, for every ε > 0. Häggkvist [10] gave a counterexample to the Erdős-Simonovits problem with chromatic number four based on the Grötzsch graph. Thomassen [15] proved that for every c > 1/3, if the minimum degree is at least cn, the chromatic number is bounded by some constant (depending on c). We completely settle the problem, describing the class of triangle-free graphs with minimum degree greater than n/3. All these graphs are four-colorable.

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