Graph Theory » Coloring » Vertex coloring

Counting 3-colorings of the hex lattice ★★

Author(s): Thomassen

Problem   Find $ \lim_{n \rightarrow \infty} (\chi( H_n , 3)) ^{ 1 / |V(H_n)| } $.

Keywords: coloring; Lieb's Ice Constant; tiling; torus

Posted by mdevos
updated July 5th, 2008
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Geometry

Dense rational distance sets in the plane ★★★

Author(s): Ulam

Problem   Does there exist a dense set $ S \subseteq {\mathbb R}^2 $ so that all pairwise distances between points in $ S $ are rational?

Keywords: integral distance; rational distance

Posted by mdevos
updated July 4th, 2008
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Graph Theory » Probabilistic G.T.

Negative association in uniform forests ★★

Author(s): Pemantle

Conjecture   Let $ G $ be a finite graph, let $ e,f \in E(G) $, and let $ F $ be the edge set of a forest chosen uniformly at random from all forests of $ G $. Then \[ {\mathbb P}(e \in F \mid f \in F}) \le {\mathbb P}(e \in F) \]

Keywords: forest; negative association

Posted by mdevos
updated June 30th, 2008
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