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

中文翻译：

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表面上的三色无三角形图III。围长图五

我们表明，至少有五个周长的4临界图的大小受其属的线性函数限制。这加强了托马森给出的此类图的大小的先前界限。它也作为描述嵌入在固定表面中的4临界无三角形图的结构的基本案例，在本系列的后续论文中将进行介绍。