Graph Theory » Coloring » Vertex coloring

Coloring squares of hypercubes ★★

Author(s): Wan

If $ G $ is a simple graph, we let $ G^{(2)} $ denote the simple graph with vertex set $ V(G) $ and two vertices adjacent if they are distance $ \le 2 $ in $ G $.

Conjecture   $ \chi(Q_d^{(2)}) = 2^{ \lfloor \log_2 d \rfloor + 1} $.

Keywords: coloring; hypercube

Posted by mdevos
updated December 10th, 2008
1 comment
Number Theory » Additive N.T.

Goldbach conjecture ★★★★

Author(s): Goldbach

Conjecture   Every even integer greater than 2 is the sum of two primes.

Keywords: additive basis; prime

Posted by Benschop
updated June 14th, 2013
3 comments
Group Theory

Growth of finitely presented groups ★★★

Author(s): Adyan

Problem   Does there exist a finitely presented group of intermediate growth?

Keywords: finitely presented; growth

Posted by mdevos
updated November 12th, 2007
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Graph Theory » Basic G.T. » Cycles

Chords of longest cycles ★★★

Author(s): Thomassen

Conjecture   If $ G $ is a 3-connected graph, every longest cycle in $ G $ has a chord.

Keywords: chord; connectivity; cycle

Posted by mdevos
updated October 14th, 2023
1 comment