Number Theory » Analytic N.T.

Are all Fermat Numbers square-free? ★★★

Author(s):

Conjecture   Are all Fermat Numbers \[ F_n = 2^{2^{n } } + 1 \] Square-Free?

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Posted by kurtulmehtap
updated August 21st, 2013
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Graph Theory » Coloring » Vertex coloring

Choosability of Graph Powers ★★

Author(s): Noel

Question  (Noel, 2013)   Does there exist a function $ f(k)=o(k^2) $ such that for every graph $ G $, \[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\]

Keywords: choosability; chromatic number; list coloring; square of a graph

Posted by Jon Noel
updated July 13th, 2013
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Graph Theory » Directed Graphs

Erdős-Posa property for long directed cycles ★★

Author(s): Havet; Maia

Conjecture   Let $ \ell \geq 2 $ be an integer. For every integer $ n\geq 0 $, there exists an integer $ t_n=t_n(\ell) $ such that for every digraph $ D $, either $ D $ has a $ n $ pairwise-disjoint directed cycles of length at least $ \ell $, or there exists a set $ T $ of at most $ t_n $ vertices such that $ D-T $ has no directed cycles of length at least $ \ell $.

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Posted by fhavet
updated June 25th, 2013
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Graph Theory » Directed Graphs

Large acyclic induced subdigraph in a planar oriented graph. ★★

Author(s): Harutyunyan

Conjecture   Every planar oriented graph $ D $ has an acyclic induced subdigraph of order at least $ \frac{3}{5} |V(D)| $.

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Posted by fhavet
updated June 25th, 2013
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