Exponentially Many Nowhere-Zero ℤ3-, ℤ4-, and ℤ6-Flows


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.

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The motivation for this research was sparked during the inspiring workshop New Trends in Graph Coloring at BIRS (Banff, Alberta) in 2016. We thank the anonymous referee for suggestions leading to a simplification of the proof of Theorem 4.5 and to an improvement of the bound.

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Corresponding authors

Correspondence to Zdeněk Dvořák or Bojan Mohar or Robert Šámal.

Additional information

Preliminary version of this work appears as extended abstract from Eurocomb 2017, published as [3].

Supported by project 14-19503S (Graph coloring and structure) of Czech Science Foundation.

Supported in part by the NSERC Discovery Grant R611450 (Canada), by the Canada Research Chairs program, and by the Research Project J1-8130 of ARRS (Slovenia).

Partially supported by grant 16-19910S of the Czech Science Foundation. Partially supported by grant LL1201 ERC CZ of the Czech Ministry of Education, Youth and Sports.

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Dvořák, Z., Mohar, B. & Šámal, R. Exponentially Many Nowhere-Zero ℤ3-, ℤ4-, and ℤ6-Flows. Combinatorica 39, 1237–1253 (2019). https://doi.org/10.1007/s00493-019-3882-x

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Mathematics Subject Classification (2010)

  • 05C21
  • 05C30