http://openproblemgarden.org/op/unit_vector_flows Comments for "Unit vector flows" en http://openproblemgarden.org/op/unit_vector_flows <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/jain">Jain</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/graph_theory">Graph Theory</a> » <a href="/category/coloring">Coloring</a> » <a href="/category/flows">Nowhere-zero flows</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; For every graph <img class="teximage" src="/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png" alt="$ G $" /> without a <a href="http://en.wikipedia.org/wiki/bridge (graph theory)">bridge</a>, there is a flow <img class="teximage" src="/files/tex/d73e0c016b36c8b5b8b2c4fec637baec205128bc.png" alt="$ \phi : E(G) \rightarrow S^2 = \{ x \in {\mathbb R}^3 : |x| = 1 \} $" />.</p> </div> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; There exists a map <img class="teximage" src="/files/tex/9e1061f676e5ae08c4736731352fdc62d6f50c6c.png" alt="$ q:S^2 \rightarrow \{-4,-3,-2,-1,1,2,3,4\} $" /> so that antipodal points of <img class="teximage" src="/files/tex/1cd459995f11529f346339e6879cf139c22ee92c.png" alt="$ S^2 $" /> receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero. </div> </tr></td> </table> </td> </tr> </table> Jain, Kamal nowhere-zero flow Graph Theory Coloring Nowhere-zero flows http://openproblemgarden.org/op/unit_vector_flows#comment Wed, 07 Mar 2007 12:08:56 +0100 mdevos 136 at http://openproblemgarden.org