If P is not equal to NP, then there is no efficient algorithm to compute either the totally cyclic or acyclic orientations of a graph. This is due to the seminal results of Jaeger, Vertigan and Welsh showing exactly which evaluations of the Tutte polynomial can be computed efficiently. We used the Tutte polynomial code due to Haggard, Pearce and myself to compute the Tutte polynomials and then just evaluated them at (0,2), (1,1) and (2,0).

No, I haven’t tested anything about the acyclic orientations. The trouble is that these conjectures about “for some c” don’t really translate to explicit small cases. I guess I could try to identify the 3-connected cubic graphs with “fewest” acyclic orientations and see where that leads.

I have a few questions related to this problem!

1. Is the link to the paper correct? Somehow I cannot access it through the link.

2. As with the number of spanning trees there is an analogous recurrence relation for computing the number of acyclic and totally cyclic orientations of a graph. We have an efficient way to compute the number of spanning trees of a graph – is the same true for he number of acyclic and totally cyclic orientations of a graph? Is there a determinant analog here as well? If not, how did you compute these quantities when testing the conjecture?

3. Have you perhaps also tested the conjecture that every 3-connected cubic graph has at most 7^{n/2}/n^c acyclic orientations?

I think just 10 vertices.

Mostly I’ve looked at the multigraphs with exactly 2n-2 edges, and I think I stopped after processing the 41 million 10-vertex 18-edge multigraphs.

But this was before I knew that a minimal counterexample must have every edge in a series or parallel class of size 2 or 3 and if I add that restriction then it would be possible to go further.

For matroids, already calculating the rank 5, size 10 matroids is a massive task.