http://openproblemgarden.org/op/universal_highly_arc_transitive_digraphs Comments for "Universal highly arc transitive digraphs" en http://openproblemgarden.org/op/universal_highly_arc_transitive_digraphs <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/cameron_peter_j">Cameron</a>; <a href="/category/praeger_cheryl_e">Praeger</a>; <a href="/category/wormald_nicholas_c">Wormald</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/graph_theory">Graph Theory</a> » <a href="/category/infinite_graphs">Infinite Graphs</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>An <em>alternating</em> walk in a digraph is a walk <img class="teximage" src="/files/tex/1be814a468eac6b1d12830d1c286f953014aa424.png" alt="$ v_0,e_1,v_1,\ldots,v_m $" /> so that the vertex <img class="teximage" src="/files/tex/9c0e2563601b47bc3e10f1194a6acf357edaffd6.png" alt="$ v_i $" /> is either the head of both <img class="teximage" src="/files/tex/c24f380acd663bc5f43554c18834ddf6f8a69584.png" alt="$ e_i $" /> and <img class="teximage" src="/files/tex/4a6533ab65f2acce6c1bbea8775ca7bcb7e5b66a.png" alt="$ e_{i+1} $" /> or the tail of both <img class="teximage" src="/files/tex/c24f380acd663bc5f43554c18834ddf6f8a69584.png" alt="$ e_i $" /> and <img class="teximage" src="/files/tex/4a6533ab65f2acce6c1bbea8775ca7bcb7e5b66a.png" alt="$ e_{i+1} $" /> for every <img class="teximage" src="/files/tex/3691086b91732fc4bd4d3060d7f3d16f8dc0b4f0.png" alt="$ 1 \le i \le m-1 $" />. A digraph is <em>universal</em> if for every pair of edges <img class="teximage" src="/files/tex/5c3827ed594e7d77374e67fdbdf4b6d801f5ec80.png" alt="$ e,f $" />, there is an alternating walk containing both <img class="teximage" src="/files/tex/5105762e0c97083905ebf07919c7d4d5ed38dce3.png" alt="$ e $" /> and <img class="teximage" src="/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png" alt="$ f $" /> </p> <div class="envtheorem"><b>Question</b>&nbsp;&nbsp; Does there exist a locally finite highly arc transitive digraph which is universal? </div> </tr></td> </table> </td> </tr> </table> Cameron, Peter J. Praeger, Cheryl E. Wormald, Nicholas C. arc transitive digraph Graph Theory Infinite Graphs http://openproblemgarden.org/op/universal_highly_arc_transitive_digraphs#comment Sun, 21 Oct 2007 03:13:49 +0200 mdevos 677 at http://openproblemgarden.org