Open Problem Garden

Frankl's union-closed sets conjecture ★★

Author(s): Frankl

Conjecture Let $F$ be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists $x$ such that $x$ is an element of at least half the members of $F$ .

Keywords:

Posted by tchow
updated December 17th, 2009

2 comments

Hamiltonicity of Cayley graphs ★★★

Author(s): Rapaport-Strasser

Question Is every Cayley graph Hamiltonian?

Keywords:

Posted by tchow
updated December 15th, 2010

4 comments

Graph Theory » Basic G.T. » Cycles

Conjecture Let $G_n$ be the bipartite graph whose vertices are the $n$ -subsets and the $(n+1)$ -subsets of a $(2n+1)$ -element set, and with inclusion as the adjacency relationship. Then $G_n$ is Hamiltonian.

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Posted by tchow
updated September 25th, 2008

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Kneser, M.

Poulsen, E. T.

pushing disks

Geometry

Kneser–Poulsen conjecture ★★★

Author(s): Kneser; Poulsen

Conjecture If a finite set of unit balls in $\mathbb{R}^n$ is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.