Number Theory » Combinatorial N.T.

Davenport's constant ★★★

Author(s):

For a finite (additive) abelian group $ G $, the Davenport constant of $ G $, denoted $ s(G) $, is the smallest integer $ t $ so that every sequence of elements of $ G $ with length $ \ge t $ has a nontrivial subsequence which sums to zero.

Conjecture   $ s( {\mathbb Z}_n^d) = d(n-1) + 1 $

Keywords: Davenport constant; subsequence sum; zero sum

Posted by mdevos
updated September 8th, 2007
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Graph Theory » Coloring

Three-chromatic (0,2)-graphs ★★

Author(s): Payan

Question   Are there any (0,2)-graphs with chromatic number exactly three?

Keywords:

Posted by Gordon Royle
updated February 6th, 2013
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Graph Theory » Topological G.T. » Drawings

Straight line representation of planar linear hypergraphs ★★

Author(s): de Fraysseix; Ossona de Mendez

Conjecture   Every planar linear hypergraph $ \mathcal H $ has a straight line representation in the plane which maps each vertex $ v $ to a point $ p(v) $ and each edge $ E $ to a straight line segment $ s(E) $, in such a way that:
    \item for each vertex $ v $ and each edge $ E $, we have: $$p(v)\in s(E)\quad\iff\quad v\in E$$ \item for each couple of distinct edges $ E_1,E_2 $, we have $$s(E_1)\cap s(E_2)=\{p(v): v\in E_1\cap E_2\}$$

Keywords: intersection graph; planar hypergraph

Posted by taxipom
updated September 4th, 2007
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