http://openproblemgarden.org/op/antichains_in_the_cycle_continuous_order Comments for "Antichains in the cycle continuous order" en http://openproblemgarden.org/op/antichains_in_the_cycle_continuous_order <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/devos">DeVos</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> <p>If <img class="teximage" src="/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png" alt="$ G $" />,<img class="teximage" src="/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png" alt="$ H $" /> are graphs, a function <img class="teximage" src="/files/tex/4fccf210ef6d182d33e76875897ce21b7ee74009.png" alt="$ f : E(G) \rightarrow E(H) $" /> is called <em>cycle-continuous</em> if the pre-image of every element of the (binary) cycle space of <img class="teximage" src="/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png" alt="$ H $" /> is a member of the cycle space of <img class="teximage" src="/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png" alt="$ G $" />.</p> <div class="envtheorem"><b>Problem</b>&nbsp;&nbsp; Does there exist an infinite set of graphs <img class="teximage" src="/files/tex/8c2ee68229cfd0ed4b4be107c721fb5ab4204c18.png" alt="$ \{G_1,G_2,\ldots \} $" /> so that there is no cycle continuous mapping between <img class="teximage" src="/files/tex/49c5f06ff437d0aa89130ae9b5fba029a520a65a.png" alt="$ G_i $" /> and <img class="teximage" src="/files/tex/a26853e93bc552976a6a2d1f0b66796c063820ae.png" alt="$ G_j $" /> whenever <img class="teximage" src="/files/tex/5c06b842f91344e3271e846fd6bff4dc9fb4ab0d.png" alt="$ i \neq j $" /> ? </div> </tr></td> </table> </td> </tr> </table> DeVos, Matt antichain cycle poset Graph Theory Coloring Nowhere-zero flows http://openproblemgarden.org/op/antichains_in_the_cycle_continuous_order#comment Sat, 12 May 2007 23:19:26 +0200 mdevos 323 at http://openproblemgarden.org