http://openproblemgarden.org/op/chromatic_number_of_frac_3_3_power_of_graph Comments for "Chromatic number of $\frac{3}{3}$-power of graph" en http://openproblemgarden.org/op/chromatic_number_of_frac_3_3_power_of_graph <table cellspacing="10"> <tr> <td> Author(s): <a href="/kpz_equation_central_limit_theorem"></a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/graph_theory">Graph Theory</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>Let <img class="teximage" src="/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png" alt="$ G $" /> be a graph and <img class="teximage" src="/files/tex/2d562ef5fbde200560d931d849894c9df0513426.png" alt="$ m,n\in \mathbb{N} $" />. The graph <img class="teximage" src="/files/tex/b971ba9ba6139f14f028cf60a78690e72c2c8b40.png" alt="$ G^{\frac{m}{n}} $" /> is defined to be the <img class="teximage" src="/files/tex/ddaab6dc091926fb1da549195000491cefae85c1.png" alt="$ m $" />-power of the <img class="teximage" src="/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png" alt="$ n $" />-subdivision of <img class="teximage" src="/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png" alt="$ G $" />. In other words, <img class="teximage" src="/files/tex/a130a1a9a495807870fa31e49ad5af1d8fb4c945.png" alt="$ G^{\frac{m}{n}}=(G^{\frac{1}{n}})^m $" />.<br /> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png" alt="$ G $" /> be a graph with <img class="teximage" src="/files/tex/25c622d02e81b413d70fba0eb49b1ca09ad2ee1b.png" alt="$ \Delta(G)\geq 2 $" />. Then <img class="teximage" src="/files/tex/77f35e23639197d1972bd2a477a135f33ef44731.png" alt="$ \chi(G^{\frac{3}{3}})\leq 2\Delta(G)+1 $" />. </div> </tr></td> </table> </td> </tr> </table> Graph Theory http://openproblemgarden.org/op/chromatic_number_of_frac_3_3_power_of_graph#comment Thu, 20 Apr 2023 14:36:32 +0200 Iradmusa 60055 at http://openproblemgarden.org