Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:37:07Z http://mathoverflow.net/feeds/question/18798 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18798/are-combinatorial-configurations-whose-levi-graphs-may-be-represented-as-covering Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines? Leah Wrenn Berman 2010-03-19T22:04:37Z 2010-03-20T17:02:30Z <p>This question is related to <a href="http://mathoverflow.net/questions/18758/drawing-a-combinatorial-3-configuration-of-points-and-lines-with-pseudolines" rel="nofollow">this previous question</a>. 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). </p> <p>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?)</p> <p>(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.)</p> http://mathoverflow.net/questions/18798/are-combinatorial-configurations-whose-levi-graphs-may-be-represented-as-covering/18858#18858 Answer by Tomaž Pisanski for Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines? Tomaž Pisanski 2010-03-20T16:55:01Z 2010-03-20T17:02:30Z <p>The fist part of your question has a negative answer, since both Fano plane (7<sub>3</sub>) and Moebius-Kantor configuration (8<sub>3</sub>) are <em>cyclic configurations</em>. <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6V00-4520GM0-F&amp;_user=10&amp;_coverDate=02%252F06%252F2002&amp;_alid=1258918945&amp;_rdoc=32&amp;_fmt=high&amp;_orig=search&amp;_cdi=5632&amp;_sort=r&amp;_docanchor=&amp;view=c&amp;_ct=39&amp;_acct=C000050221&amp;_version=1&amp;_urlVersion=0&amp;_userid=10&amp;md5=966cc8a16ea0603a5f1aa4ac9044ccd2" rel="nofollow">Here</a> 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 <em>dipole</em> 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.</p> <p>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<sub>w</sub> to produce a straight-line drawing of the corresponding <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6WDY-48BKR3H-6&amp;_user=10&amp;_coverDate=05%252F31%252F2003&amp;_alid=1258918945&amp;_rdoc=35&amp;_fmt=high&amp;_orig=search&amp;_cdi=6779&amp;_sort=r&amp;_docanchor=&amp;view=c&amp;_ct=39&amp;_acct=C000050221&amp;_version=1&amp;_urlVersion=0&amp;_userid=10&amp;md5=144ff4ac2004d0e709e163b00381794d" rel="nofollow">polycyclic configuration</a> (v<sub>3</sub>). Unfortunately this method does not apply to cyclic configurations (v<sub>k</sub>), 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 <em>rotational symmetry</em>. </p>