We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the coloring of a bounded number of vertices.

中文翻译：

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VII. 曲面上的三色无三角形图。线性时间算法

我们给出了一个线性时间算法来确定嵌入在固定表面中的无三角形图的 3 着色性，以及在肯定情况下输出 3 着色的二次时间算法。该算法还允许规定有限数量的顶点的着色。