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Graph Theory » Directed Graphs

Seymour's Second Neighbourhood Conjecture ★★★

Author(s): Seymour

Conjecture   Any oriented graph has a vertex whose outdegree is at most its second outdegree.

Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour

Posted by nkorppi
updated March 18th, 2011
5 comments
Group Theory

Which lattices occur as intervals in subgroup lattices of finite groups? ★★★★

Author(s):

Conjecture  

There exists a finite lattice that is not an interval in the subgroup lattice of a finite group.

Keywords: congruence lattice; finite groups

Posted by williamdemeo
updated February 22nd, 2011
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Number Theory

Quartic rationally derived polynomials ★★★

Author(s): Buchholz; MacDougall

Call a polynomial $ p \in {\mathbb Q}[x] $ rationally derived if all roots of $ p $ and the nonzero derivatives of $ p $ are rational.

Conjecture   There does not exist a quartic rationally derived polynomial $ p \in {\mathbb Q}[x] $ with four distinct roots.

Keywords: derivative; diophantine; elliptic; polynomial

Posted by mdevos
updated February 16th, 2011
1 comment
Topology

Nonseparating planar continuum ★★

Author(s):

Conjecture   Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?

A set has the fixed point property if every continuous map from it into itself has a fixed point.

Keywords: fixed point

Posted by porton
updated February 16th, 2011
1 comment
Topology

Hilbert-Smith conjecture ★★

Author(s): David Hilbert; Paul A. Smith

Conjecture   Let $ G $ be a locally compact topological group. If $ G $ has a continuous faithful group action on an $ n $-manifold, then $ G $ is a Lie group.

Keywords:

Posted by porton
updated February 6th, 2011
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Algebra

trace inequality ★★

Author(s):

Let $ A,B $ be positive semidefinite, by Jensen's inequality, it is easy to see $ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $, whenever $ s>r>0 $.

What about the $ tr(A^s+B^s)^{\frac{1}{s}}\leq tr(A^r+B^r)^{\frac{1}{r}} $, is it still valid?

Keywords:

Posted by Miwa Lin
updated December 20th, 2010
3 comments
Graph Theory » Coloring » Nowhere-zero flows

Real roots of the flow polynomial ★★

Author(s): Welsh

Conjecture   All real roots of nonzero flow polynomials are at most 4.

Keywords: flow polynomial; nowhere-zero flow

Posted by mdevos
updated December 17th, 2010
2 comments
Graph Theory » Basic G.T. » Cycles

Hamiltonicity of Cayley graphs ★★★

Author(s): Rapaport-Strasser

Question   Is every Cayley graph Hamiltonian?

Keywords:

Posted by tchow
updated December 15th, 2010
4 comments
Algebra

Finite Lattice Representation Problem ★★★★

Author(s):

Conjecture  

There exists a finite lattice which is not the congruence lattice of a finite algebra.

Keywords: congruence lattice; finite algebra

Posted by williamdemeo
updated December 10th, 2010
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Topology

Outer reloid of restricted funcoid ★★

Author(s): Porton

Question   $ ( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f) \cap^{\mathsf{RLD}} ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) $ for every filter objects $ \mathcal{A} $ and $ \mathcal{B} $ and a funcoid $ f\in\mathsf{FCD}(\mathrm{Src}\,f; \mathrm{Dst}\,f) $?

Keywords: direct product of filters; outer reloid

Posted by porton
updated December 3rd, 2010
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Graph Theory

Star chromatic index of complete graphs ★★

Author(s): Dvorak; Mohar; Samal

Conjecture   Is it possible to color edges of the complete graph $ K_n $ using $ O(n) $ colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?

Equivalently: is the star chromatic index of $ K_n $ linear in $ n $?

Keywords: complete graph; edge coloring; star coloring

Posted by Robert Samal
updated November 16th, 2010
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Graph Theory

Star chromatic index of cubic graphs ★★

Author(s): Dvorak; Mohar; Samal

The star chromatic index $ \chi_s'(G) $ of a graph $ G $ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.

Question   Is it true that for every (sub)cubic graph $ G $, we have $ \chi_s'(G) \le 6 $?

Keywords: edge coloring; star coloring

Posted by Robert Samal
updated November 16th, 2010
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Number Theory » Analytic N.T.

Lindelöf hypothesis ★★

Author(s): Lindelöf

Conjecture   For any $ \epsilon>0 $ $$\zeta\left(\frac12 + it\right) \mbox{ is }\mathcal{O}(t^\epsilon).$$

Since $ \epsilon $ can be replaced by a smaller value, we can also write the conjecture as, for any positive $ \epsilon $, $$\zeta\left(\frac12 + it\right) \mbox{ is }o(t^\varepsilon).$$

Keywords: Riemann Hypothesis; zeta

Posted by porton
updated September 20th, 2010
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Logic

Termination of the sixth Goodstein Sequence ★

Author(s): Graham

Question   How many steps does it take the sixth Goodstein sequence to terminate?

Keywords: Goodstein Sequence

Posted by mdevos
updated September 17th, 2010
5 comments
Graph Theory » Topological G.T. » Genus

Consecutive non-orientable embedding obstructions ★★★

Author(s):

Conjecture   Is there a graph $ G $ that is a minor-minimal obstruction for two non-orientable surfaces?

Keywords: minor; surface

Posted by Bruce Richter
updated September 15th, 2010
2 comments
Combinatorics » Ramsey Theory

Diagonal Ramsey numbers ★★★★

Author(s): Erdos

Let $ R(k,k) $ denote the $ k^{th} $ diagonal Ramsey number.

Conjecture   $ \lim_{k \rightarrow \infty} R(k,k) ^{\frac{1}{k}} $ exists.
Problem   Determine the limit in the above conjecture (assuming it exists).

Keywords: Ramsey number

Posted by mdevos
updated September 13th, 2010
2 comments
Topology

The 4x5 chessboard complex is the complement of a link, which link? ★★

Author(s): David Eppstein

Problem   Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.

Keywords: knot theory, links, chessboard complex

Posted by rybu
updated September 4th, 2010
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Algebra

Elementary symmetric of a sum of matrices ★★★

Author(s):

Problem  

Given a Matrix $ A $, the $ k $-th elementary symmetric function of $ A $, namely $ S_k(A) $, is defined as the sum of all $ k $-by-$ k $ principal minors.

Find a closed expression for the $ k $-th elementary symmetric function of a sum of N $ n $-by-$ n $ matrices, with $ 0\le N\le k\le n $ by using partitions.

Keywords:

Posted by rscosa
updated August 27th, 2010
1 comment
Geometry

Monochromatic empty triangles ★★★

Author(s):

If $ X \subseteq {\mathbb R}^2 $ is a finite set of points which is 2-colored, an empty triangle is a set $ T \subseteq X $ with $ |T|=3 $ so that the convex hull of $ T $ is disjoint from $ X \setminus T $. We say that $ T $ is monochromatic if all points in $ T $ are the same color.

Conjecture   There exists a fixed constant $ c $ with the following property. If $ X \subseteq {\mathbb R}^2 $ is a set of $ n $ points in general position which is 2-colored, then it has $ \ge cn^2 $ monochromatic empty triangles.

Keywords: empty triangle; general position; ramsey theory

Posted by mdevos
updated August 27th, 2010
4 comments
Combinatorics » Ramsey Theory

Edge-antipodal colorings of cubes ★★

Author(s): Norine

We let $ Q_d $ denote the $ d $-dimensional cube graph. A map $ \phi : E(Q_d) \rightarrow \{0,1\} $ is called edge-antipodal if $ \phi(e) \neq \phi(e') $ whenever $ e,e' $ are antipodal edges.

Conjecture   If $ d \ge 2 $ and $ \phi : E(Q_d) \rightarrow \{0,1\} $ is edge-antipodal, then there exist a pair of antipodal vertices $ v,v' \in V(Q_d) $ which are joined by a monochromatic path.

Keywords: antipodal; cube; edge-coloring

Posted by mdevos
updated August 20th, 2010
5 comments