For a stronger result on the enumeration of -flows with a bound on the maximum flow amount per edge, again using Robbins' theorem on totally cyclic orientations, see Theorem 2 of Kochol, Martin (2002), "Polynomials associated with nowhere-zero flows", Journal of Combinatorial Theory, Series B, 84 (2): 260–269, doi:10.1006/jctb.2001.2081, MR1889258
doi.org
Tutte, W. T. (1954). "A contribution to the theory of chromatic polynomials". Canadian Journal of Mathematics. 6: 80–91. doi:10.4153/CJM-1954-010-9.
For a stronger result on the enumeration of -flows with a bound on the maximum flow amount per edge, again using Robbins' theorem on totally cyclic orientations, see Theorem 2 of Kochol, Martin (2002), "Polynomials associated with nowhere-zero flows", Journal of Combinatorial Theory, Series B, 84 (2): 260–269, doi:10.1006/jctb.2001.2081, MR1889258