http://openproblemgarden.org/op/drawing_disconnected_graphs_on_surfaces Comments for "Drawing disconnected graphs on surfaces" en http://openproblemgarden.org/op/drawing_disconnected_graphs_on_surfaces#comment-225 <p>Hello,</p> <p>You don't mention reference in this problem, though it is said that some work has been made. Would it be possible to know how the projective plane has been shown to verify this conjecture ?</p> <p>Thanks in advance for any piece of information.</p> <p>Laurent Beaudou</p> Sun, 30 Sep 2007 16:29:46 +0200 lbeaudou comment 225 at http://openproblemgarden.org http://openproblemgarden.org/op/drawing_disconnected_graphs_on_surfaces <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/devos">DeVos</a>; <a href="/category/mohar">Mohar</a>; <a href="/category/samal">Samal</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/graph_theory">Graph Theory</a> » <a href="/category/topological_graph_theory">Topological G.T.</a> » <a href="/category/crossing_numbers">Crossing numbers</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png" alt="$ G $" /> be the disjoint union of the graphs <img class="teximage" src="/files/tex/1475475906abb943f311289729184c527071d32f.png" alt="$ G_1 $" /> and <img class="teximage" src="/files/tex/7c390ecd91deb4948e85912eba8f5594f03ea2f4.png" alt="$ G_2 $" /> and let <img class="teximage" src="/files/tex/70e73d02e8dfe5c2df3217d0bb04993df8dd0712.png" alt="$ \Sigma $" /> be a surface. Is it true that every optimal drawing of <img class="teximage" src="/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png" alt="$ G $" /> on <img class="teximage" src="/files/tex/70e73d02e8dfe5c2df3217d0bb04993df8dd0712.png" alt="$ \Sigma $" /> has the property that <img class="teximage" src="/files/tex/1475475906abb943f311289729184c527071d32f.png" alt="$ G_1 $" /> and <img class="teximage" src="/files/tex/7c390ecd91deb4948e85912eba8f5594f03ea2f4.png" alt="$ G_2 $" /> are disjoint? </div> </tr></td> </table> </td> </tr> </table> DeVos, Matt Mohar, Bojan Samal, Robert crossing number surface Graph Theory Topological Graph Theory Crossing numbers http://openproblemgarden.org/op/drawing_disconnected_graphs_on_surfaces#comment Sat, 12 May 2007 22:27:10 +0200 mdevos 322 at http://openproblemgarden.org