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Number Theory » Analytic N.T.

Distribution and upper bound of mimic numbers ★★

Author(s): Bhattacharyya

Problem  

Let the notation $ a|b $ denote ''$ a $ divides $ b $''. The mimic function in number theory is defined as follows [1].

Definition   For any positive integer $ \mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i} $ divisible by $ \mathcal{D} $, the mimic function, $ f(\mathcal{D} | \mathcal{N}) $, is given by,

$$ f(\mathcal{D} | \mathcal{N}) = \sum_{i=0}^{n}\mathcal{X}_{i}(\mathcal{M}-\mathcal{D})^{i} $$

By using this definition of mimic function, the mimic number of any non-prime integer is defined as follows [1].

Definition   The number $ m $ is defined to be the mimic number of any positive integer $ \mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i} $, with respect to $ \mathcal{D} $, for the minimum value of which $ f^{m}(\mathcal{D} | \mathcal{N}) = \mathcal{D} $.

Given these two definitions and a positive integer $ \mathcal{D} $, find the distribution of mimic numbers of those numbers divisible by $ \mathcal{D} $.

Again, find whether there is an upper bound of mimic numbers for a set of numbers divisible by any fixed positive integer $ \mathcal{D} $.

Keywords: Divisibility; mimic function; mimic number

Posted by facility_cttb@i...
updated June 20th, 2009
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Graph Theory » Probabilistic G.T.

Coloring random subgraphs ★★

Author(s): Bukh

If $ G $ is a graph and $ p \in [0,1] $, we let $ G_p $ denote a subgraph of $ G $ where each edge of $ G $ appears in $ G_p $ with independently with probability $ p $.

Problem   Does there exist a constant $ c $ so that $ {\mathbb E}(\chi(G_{1/2})) > c \frac{\chi(G)}{\log \chi(G)} $?

Keywords: coloring; random graph

Posted by mdevos
updated June 4th, 2009
1 comment
Graph Theory » Coloring » Vertex coloring

Are vertex minor closed classes chi-bounded? ★★

Author(s): Geelen

Question   Is every proper vertex-minor closed class of graphs chi-bounded?

Keywords: chi-bounded; circle graph; coloring; vertex minor

Posted by mdevos
updated May 16th, 2009
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Graph Theory » Coloring » Vertex coloring

Graphs with a forbidden induced tree are chi-bounded ★★★

Author(s): Gyarfas

Say that a family $ {\mathcal F} $ of graphs is $ \chi $-bounded if there exists a function $ f: {\mathbb N} \rightarrow {\mathbb N} $ so that every $ G \in {\mathcal F} $ satisfies $ \chi(G) \le f (\omega(G)) $.

Conjecture   For every fixed tree $ T $, the family of graphs with no induced subgraph isomorphic to $ T $ is $ \chi $-bounded.

Keywords: chi-bounded; coloring; excluded subgraph; tree

Posted by mdevos
updated May 16th, 2009
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Graph Theory » Basic G.T.

Asymptotic Distribution of Form of Polyhedra ★★

Author(s): Rüdinger

Problem   Consider the set of all topologically inequivalent polyhedra with $ k $ edges. Define a form parameter for a polyhedron as $ \beta:= v/(k+2) $ where $ v $ is the number of vertices. What is the distribution of $ \beta $ for $ k \to \infty $?

Keywords: polyhedral graphs, distribution

Posted by andreasruedinger
updated May 9th, 2009
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Graph Theory » Topological G.T.

Domination in plane triangulations ★★

Author(s): Matheson; Tarjan

Conjecture   Every sufficiently large plane triangulation $ G $ has a dominating set of size $ \le \frac{1}{4} |V(G)| $.

Keywords: coloring; domination; multigrid; planar graph; triangulation

Posted by mdevos
updated May 4th, 2009
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Geometry

Erdös-Szekeres conjecture ★★★

Author(s): Erdos; Szekeres

Conjecture   Every set of $ 2^{n-2} + 1 $ points in the plane in general position contains a subset of $ n $ points which form a convex $ n $-gon.

Keywords: combinatorial geometry; Convex Polygons; ramsey theory

Posted by mdevos
updated April 14th, 2009
1 comment
Graph Theory » Coloring » Nowhere-zero flows

4-flow conjecture ★★★

Author(s): Tutte

Conjecture   Every bridgeless graph with no Petersen minor has a nowhere-zero 4-flow.

Keywords: minor; nowhere-zero flow; Petersen graph

Posted by mdevos
updated April 12th, 2009
2 comments
Geometry

Inequality of the means ★★★

Author(s):

Question   Is is possible to pack $ n^n $ rectangular $ n $-dimensional boxes each of which has side lengths $ a_1,a_2,\ldots,a_n $ inside an $ n $-dimensional cube with side length $ a_1 + a_2 + \ldots a_n $?

Keywords: arithmetic mean; geometric mean; Inequality; packing

Posted by mdevos
updated April 6th, 2009
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Theoretical Comp. Sci. » Complexity

P vs. PSPACE ★★★

Author(s): Folklore

Problem   Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?

Keywords: P; PSPACE; separation; unconditional

Posted by cwenner
updated April 4th, 2009
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Theoretical Comp. Sci.

Sums of independent random variables with unbounded variance ★★

Author(s): Feige

Conjecture   If $ X_1, \dotsc, X_n \geq 0 $ are independent random variables with $ \mathbb{E}[X_i] \leq \mu $, then $$\mathrm{Pr} \left( \sum X_i - \mathbb{E} \left[ \sum X_i \right ] < \delta \mu \right) \geq \min \left ( (1 + \delta)^{-1} \delta, e^{-1} \right).$$

Keywords: Inequality; Probability Theory; randomness in TCS

Posted by cwenner
updated April 2nd, 2009
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Graph Theory » Topological G.T. » Coloring

Grunbaum's Conjecture ★★★

Author(s): Grunbaum

Conjecture   If $ G $ is a simple loopless triangulation of an orientable surface, then the dual of $ G $ is 3-edge-colorable.

Keywords: coloring; surface

Posted by mdevos
updated March 23rd, 2009
5 comments
Theoretical Comp. Sci. » Complexity » Hardness of Approximation

Refuting random 3SAT-instances on $O(n)$ clauses (weak form) ★★★

Author(s): Feige

Conjecture   For every rational $ \epsilon > 0 $ and every rational $ \Delta $, there is no polynomial-time algorithm for the following problem.

Given is a 3SAT (3CNF) formula $ I $ on $ n $ variables, for some $ n $, and $ m = \floor{\Delta n} $ clauses drawn uniformly at random from the set of formulas on $ n $ variables. Return with probability at least 0.5 (over the instances) that $ I $ is typical without returning typical for any instance with at least $ (1 - \epsilon)m $ simultaneously satisfiable clauses.

Keywords: NP; randomness in TCS; satisfiability

Posted by cwenner
updated February 27th, 2009
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Graph Theory » Algebraic G.T.

Does the chromatic symmetric function distinguish between trees? ★★

Author(s): Stanley

Problem   Do there exist non-isomorphic trees which have the same chromatic symmetric function?

Keywords: chromatic polynomial; symmetric function; tree

Posted by mdevos
updated February 25th, 2009
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Graph Theory

Shannon capacity of the seven-cycle ★★★

Author(s):

Problem   What is the Shannon capacity of $ C_7 $?

Keywords:

Posted by tchow
updated February 19th, 2009
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Number Theory » Combinatorial N.T.

Frobenius number of four or more integers ★★

Author(s):

Problem   Find an explicit formula for Frobenius number $ g(a_1, a_2, \dots, a_n) $ of co-prime positive integers $ a_1, a_2, \dots, a_n $ for $ n\geq 4 $.

Keywords:

Posted by maxal
updated November 26th, 2008
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Number Theory » Computational N.T.

Magic square of squares ★★

Author(s): LaBar

Question   Does there exist a $ 3\times 3 $ magic square composed of distinct perfect squares?

Keywords:

Posted by maxal
updated November 23rd, 2008
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Group Theory

Inverse Galois Problem ★★★★

Author(s): Hilbert

Conjecture   Every finite group is the Galois group of some finite algebraic extension of $ \mathbb Q $.

Keywords:

Posted by tchow
updated October 13th, 2008
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Graph Theory » Coloring » Edge coloring

Seymour's r-graph conjecture ★★★

Author(s): Seymour

An $ r $-graph is an $ r $-regular graph $ G $ with the property that $ |\delta(X)| \ge r $ for every $ X \subseteq V(G) $ with odd size.

Conjecture   $ \chi'(G) \le r+1 $ for every $ r $-graph $ G $.

Keywords: edge-coloring; r-graph

Posted by mdevos
updated October 3rd, 2008
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Graph Theory » Coloring » Edge coloring

Edge list coloring conjecture ★★★

Author(s):

Conjecture   Let $ G $ be a loopless multigraph. Then the edge chromatic number of $ G $ equals the list edge chromatic number of $ G $.

Keywords:

Posted by tchow
updated September 25th, 2008
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