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We show that the size of a 4-critical graph of girth at least five is boundedby a linear function of its genus. This strengthens the previous bound on thesize of such graphs given by Thomassen. It also serves as the basic case forthe description of the structure of 4-critical triangle-free graphs embedded ina fixed surface, presented in a future paper of this series.

Three-coloring triangle-free graphs on surfaces III. Graphs of girth five