Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines? - MathOverflow

Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?

2010-03-19T22:04:37Z

This question is related to this previous question. Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in a number of such cases, it is possible to represent the combinatorial configuration as a geometric configuration (i.e., using points and straight lines in the Euclidean plane).

Given a bipartite graph which is obtained from a voltage graph, we can view it as a Levi graph of some combinatorial configuration. Is it possible to draw all such configurations using pseudolines? If not, are there easy/known constraints on the ones that fail? (e.g., if there are more than x points in the configuration, then things work? You can't use such-and-so groups as the cyclic group for the voltage graph?)

(Does the Heawood graph have a voltage-graph representation? If so, it makes the first question easy to answer, but the second one is still interesting. Maybe.)

Answer by Tomaž Pisanski for Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?

2010-03-20T16:55:01Z

The fist part of your question has a negative answer, since both Fano plane (7₃) and Moebius-Kantor configuration (8₃) are cyclic configurations. Here it is shown that the cyclic covering graphs over a dipole with girth at least 6 are exactly Levi graphs of combinatorial cyclic configurations. In your terminology, each Levi graph of a cyclic configuration has a "voltage-graph representation". Note: a dipole is a graph consisting of two vertices and a number of parallel edges between them. In particular both the Heawood graph and the Moebius-Kantor graph are counterexaples.

If v = uw is a composite number, it is sometimes possible to find a suitable voltage graph on 2u vertices and voltages from the cyclic group Z_w to produce a straight-line drawing of the corresponding polycyclic configuration (v₃). Unfortunately this method does not apply to cyclic configurations (v_k), for v prime and is not certainly not understood well for k > 4. I am not sure what happens if pseudolines are admitted in such drawings having rotational symmetry.