We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 “templates” (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not 3-colorable if and only if it contains a subgraph matching one of the templates. As a consequence, we show every triangle-free graph drawn in the torus with edge-width at least six is 3-colorable, a key property used in an efficient 3-colorability algorithm for triangle-free toroidal graphs.

中文翻译：

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4-临界无三角形环形图的表征

我们以 186 个“模板”（某些面由任意四边形填充的图）的形式给出了在环面中绘制的无三角形图的 3 着色性的精确表征，因此如果该类中的图不是 3 着色的并且仅当它包含与其中一个模板匹配的子图时。因此，我们展示了在环面中绘制的每个边宽至少为 6 的无三角形图都是 3 色的，这是用于无三角形环面图的有效 3 色算法中的关键属性。