We prove that, in several settings, a graph has exponentially many nowhere-zero flows. These results may be seen as a counting alternative to the well-known proofs of existence of ℤ3-, ℤ4-, and ℤ6-flows. In the dual setting, proving exponential number of 3-colorings of planar triangle-free graphs is a related open question due to Thomassen.

中文翻译：

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成倍数的无处-零 ℤ3-、ℤ4- 和 ℤ6-流

我们证明，在多种设置中，一个图具有成倍数的无处零流。这些结果可以看作是众所周知的ℤ3-、ℤ4-和ℤ6-流存在证明的计数替代方案。在对偶设置中，证明无平面三角形图的 3-着色指数数是 Thomassen 的一个相关的未决问题。