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Lucas Numbers Modulo m ★★

Author(s):

Conjecture   The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the following: 2, 4, 6, 7, 14, 3^k, k >=1.

Keywords: Lucas numbers

SimCity BuildIt Generator Cheats Unlimited Resources No Jailbreak (Premium Orginal Generator) ★★

Author(s):

SimCity BuildIt Generator Cheats Unlimited Resources No Jailbreak (Premium Orginal Generator)

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Divisibility of central binomial coefficients ★★

Author(s): Graham

Problem  (1)   Prove that there exist infinitely many positive integers $ n $ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$
Problem  (2)   Prove that there exists only a finite number of positive integers $ n $ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7\cdot 11) = 1.$$

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Point sets with no empty pentagon

Author(s): Wood

Problem   Classify the point sets with no empty pentagon.

Keywords: combinatorial geometry; visibility graph

Large induced forest in a planar graph. ★★

Author(s): Abertson; Berman

Conjecture   Every planar graph on $ n $ verices has an induced forest with at least $ n/2 $ vertices.

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Raid Shadow Legends Cheats Generator Working (refreshed version) ★★

Author(s):

Raid Shadow Legends Cheats Generator Working (refreshed version)

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Inverse Galois Problem ★★★★

Author(s): Hilbert

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

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Hamilton cycle in small d-diregular graphs ★★

Author(s): Jackson

An directed graph is $ k $-diregular if every vertex has indegree and outdegree at least $ k $.

Conjecture   For $ d >2 $, every $ d $-diregular oriented graph on at most $ 4d+1 $ vertices has a Hamilton cycle.

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Perfect cuboid ★★

Author(s):

Conjecture   Does a perfect cuboid exist?

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Hedetniemi's Conjecture ★★★

Author(s): Hedetniemi

Conjecture   If $ G,H $ are simple finite graphs, then $ \chi(G \times H) = \min \{ \chi(G), \chi(H) \} $.

Here $ G \times H $ is the tensor product (also called the direct or categorical product) of $ G $ and $ H $.

Keywords: categorical product; coloring; homomorphism; tensor product

Free Matchington Mansion Cheats Stars Coins Generator 2024 (Legal) ★★

Author(s):

Free Matchington Mansion Cheats Stars Coins Generator 2024 (Legal)

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My Singing Monsters Cheats Generator Android Ios 2024 Cheats Generator (re-designed) ★★

Author(s):

My Singing Monsters Cheats Generator Android Ios 2024 Cheats Generator (re-designed)

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Vertex Coloring of graph fractional powers ★★★

Author(s): Iradmusa

Conjecture   Let $ G $ be a graph and $ k $ be a positive integer. The $ k- $power of $ G $, denoted by $ G^k $, is defined on the vertex set $ V(G) $, by connecting any two distinct vertices $ x $ and $ y $ with distance at most $ k $. In other words, $ E(G^k)=\{xy:1\leq d_G(x,y)\leq k\} $. Also $ k- $subdivision of $ G $, denoted by $ G^\frac{1}{k} $, is constructed by replacing each edge $ ij $ of $ G $ with a path of length $ k $. Note that for $ k=1 $, we have $ G^\frac{1}{1}=G^1=G $.
Now we can define the fractional power of a graph as follows:
Let $ G $ be a graph and $ m,n\in \mathbb{N} $. The graph $ G^{\frac{m}{n}} $ is defined by the $ m- $power of the $ n- $subdivision of $ G $. In other words $ G^{\frac{m}{n}}\isdef (G^{\frac{1}{n}})^m $.
Conjecture. Let $ G $ be a connected graph with $ \Delta(G)\geq3 $ and $ m $ be a positive integer greater than 1. Then for any positive integer $ n>m $, we have $ \chi(G^{\frac{m}{n}})=\omega(G^\frac{m}{n}) $.
In [1], it was shown that this conjecture is true in some special cases.

Keywords: chromatic number, fractional power of graph, clique number

Free Warframe Cheats Platinum Generator 2024 (Legal) ★★

Author(s):

Free Warframe Cheats Platinum Generator 2024 (Legal)

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Directed path of length twice the minimum outdegree ★★★

Author(s): Thomassé

Conjecture   Every oriented graph with minimum outdegree $ k $ contains a directed path of length $ 2k $.

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Exact colorings of graphs ★★

Author(s): Erickson

Conjecture   For $ c \geq m \geq 1 $, let $ P(c,m) $ be the statement that given any exact $ c $-coloring of the edges of a complete countably infinite graph (that is, a coloring with $ c $ colors all of which must be used at least once), there exists an exactly $ m $-colored countably infinite complete subgraph. Then $ P(c,m) $ is true if and only if $ m=1 $, $ m=2 $, or $ c=m $.

Keywords: graph coloring; ramsey theory

Cookie Run Kingdom Cheats Generator (New Working Cheats Generator 2024) ★★

Author(s):

Cookie Run Kingdom Cheats Generator (New Working Cheats Generator 2024)

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The Borodin-Kostochka Conjecture ★★

Author(s): Borodin; Kostochka

Conjecture   Every graph with maximum degree $ \Delta \geq 9 $ has chromatic number at most $ \max\{\Delta-1, \omega\} $.

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Length of surreal product

Author(s): Gonshor

Conjecture   Every surreal number has a unique sign expansion, i.e. function $ f: o\rightarrow \{-, +\} $, where $ o $ is some ordinal. This $ o $ is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $ s $ as $ \ell(s) $.

It is easy to prove that

$$ \ell(s+t) \leq \ell(s)+\ell(t) $$

What about

$$ \ell(s\times t) \leq \ell(s)\times\ell(t) $$

?

Keywords: surreal numbers

Jurassic World Alive Coins Cash Cheats 2024 Update Cheat (Verified) ★★

Author(s):

Jurassic World Alive Coins Cash Cheats 2024 Update Cheat (Verified)

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The three 4-flows conjecture ★★

Author(s): DeVos

Conjecture   For every graph $ G $ with no bridge, there exist three disjoint sets $ A_1,A_2,A_3 \subseteq E(G) $ with $ A_1 \cup A_2 \cup A_3 = E(G) $ so that $ G \setminus A_i $ has a nowhere-zero 4-flow for $ 1 \le i \le 3 $.

Keywords: nowhere-zero flow

inverse of an integer matrix ★★

Author(s): Gregory

Question   I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all $ \ge 2 $. Suppose X is an m-by-n integer matrix $ (m \le n) $. Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.

Keywords: invertable matrices, integer matrices

Latest Bingo Blitz Cheats Generator 999K Credits Free 2024 in 5 minutes (Up To) ★★

Author(s):

Latest Bingo Blitz Cheats Generator 999K Credits Free 2024 in 5 minutes (Up To)

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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

New.updated Super Meat Boy Forever Points Cheats 2024 Free No Verification "Free" ★★

Author(s):

New.updated Super Meat Boy Forever Points Cheats 2024 Free No Verification "Free"

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Dragon Ball Z Dokkan Battle Cheats Generator 2024 Update (FREE) ★★

Author(s):

Dragon Ball Z Dokkan Battle Cheats Generator 2024 Update (FREE)

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Antichains in the cycle continuous order ★★

Author(s): DeVos

If $ G $,$ H $ are graphs, a function $ f : E(G) \rightarrow E(H) $ is called cycle-continuous if the pre-image of every element of the (binary) cycle space of $ H $ is a member of the cycle space of $ G $.

Problem   Does there exist an infinite set of graphs $ \{G_1,G_2,\ldots \} $ so that there is no cycle continuous mapping between $ G_i $ and $ G_j $ whenever $ i \neq j $ ?

Keywords: antichain; cycle; poset

Melnikov's valency-variety problem

Author(s): Melnikov

Problem   The valency-variety $ w(G) $ of a graph $ G $ is the number of different degrees in $ G $. Is the chromatic number of any graph $ G $ with at least two vertices greater than $$\ceil{ \frac{\floor{w(G)/2}}{|V(G)| - w(G)} } ~ ?$$

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Clash of Clans Gems Cheats 2023-2024 Edition Hacks (NEW-FREE!!) ★★

Author(s):

Clash of Clans Gems Cheats 2023-2024 Edition Hacks (NEW-FREE!!)

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Infinite uniquely hamiltonian graphs ★★

Author(s): Mohar

Problem   Are there any uniquely hamiltonian locally finite 1-ended graphs which are regular of degree $ r > 2 $?

Keywords: hamiltonian; infinite graph; uniquely hamiltonian

Yu Gi Oh Duel Links Cheats Generator 2024 (No Human Verification) ★★

Author(s):

Yu Gi Oh Duel Links Cheats Generator 2024 (No Human Verification)

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Slice-ribbon problem ★★★★

Author(s): Fox

Conjecture   Given a knot in $ S^3 $ which is slice, is it a ribbon knot?

Keywords: cobordism; knot; ribbon; slice

Linear Hypergraphs with Dimension 3 ★★

Author(s): de Fraysseix; Ossona de Mendez; Rosenstiehl

Conjecture   Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.

Keywords: Hypergraphs

My Singing Monsters Cheats Generator 2024 (rejuvenated Generator) ★★

Author(s):

My Singing Monsters Cheats Generator 2024 (rejuvenated Generator)

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"Working Cheats" Subway Surfers Coins Keys Generator Ios Android 2024 ★★

Author(s):

"Working Cheats" Subway Surfers Coins Keys Generator Ios Android 2024

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War Dragons Rubies Cheats 2024 (rejuvenated cheats) ★★

Author(s):

War Dragons Rubies Cheats 2024 (rejuvenated cheats)

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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

Algorithm for graph homomorphisms ★★

Author(s): Fomin; Heggernes; Kratsch

Question  

Is there an algorithm that decides, for input graphs $ G $ and $ H $, whether there exists a homomorphism from $ G $ to $ H $ in time $ O(c^{|V(G)|+|V(H)|}) $ for some constant $ c $?

Keywords: algorithm; Exponential-time algorithm; homomorphism

Sky Children of the Light Unlimited Candle Cheats (New 2024) ★★

Author(s):

Sky Children of the Light Unlimited Candle Cheats (New 2024)

Keywords:

Convex Equipartitions with Extreme Perimeter ★★

Author(s): Nandakumar

To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.

Remark: It appears maximizing the total perimeter is the easier problem.

Keywords: convex equipartition

Unions of triangle free graphs ★★★

Author(s): Erdos; Hajnal

Problem   Does there exist a graph with no subgraph isomorphic to $ K_4 $ which cannot be expressed as a union of $ \aleph_0 $ triangle free graphs?

Keywords: forbidden subgraph; infinite graph; triangle free

Closing Lemma for Diffeomorphism (Dynamical Systems) ★★★★

Author(s): Charles Pugh

Conjecture   Let $ f\in Diff^{r}(M) $ and $ p\in\omega_{f}  $. Then for any neighborhood $ V_{f}\subset Diff^{r}(M)  $ there is $ g\in V_{f} $ such that $ p $ is periodic point of $ g $

There is an analogous conjecture for flows ( $ C^{r} $ vector fields . In the case of diffeos this was proved by Charles Pugh for $ r = 1 $. In the case of Flows this has been solved by Sushei Hayahshy for $ r = 1 $ . But in the two cases the problem is wide open for $ r > 1 $

Keywords: Dynamics , Pertubation

Free Hollywood Story Free Diamonds Gems Cheats 2024 (Safe) ★★

Author(s):

Free Hollywood Story Free Diamonds Gems Cheats 2024 (Safe)

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Rise Of Kingdoms Cheats Generator 2024-2024 (NEW-FREE!!) ★★

Author(s):

Rise Of Kingdoms Cheats Generator 2024-2024 (NEW-FREE!!)

Keywords:

Raid Shadow Legends Generator Cheats Free 2024 in 5 minutes (New Generator Cheats Raid Shadow Legends) ★★

Author(s):

Raid Shadow Legends Generator Cheats Free 2024 in 5 minutes (New Generator Cheats Raid Shadow Legends)

Keywords:

Match Masters Coins Cheats 2024 Update (FREE!!) ★★

Author(s):

Match Masters Coins Cheats 2024 Update (FREE!!)

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The Alon-Tarsi basis conjecture ★★

Author(s): Alon; Linial; Meshulam

Conjecture   If $ B_1,B_2,\ldots B_p $ are invertible $ n \times n $ matrices with entries in $ {\mathbb Z}_p $ for a prime $ p $, then there is a $ n \times (p-1)n $ submatrix $ A $ of $ [B_1 B_2 \ldots B_p] $ so that $ A $ is an AT-base.

Keywords: additive basis; matrix

The Sims Mobile Cheats Generator Working Android Ios 2024 Cheats Generator (Newly Discovered) ★★

Author(s):

The Sims Mobile Cheats Generator Working Android Ios 2024 Cheats Generator (Newly Discovered)

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Diophantine quintuple conjecture ★★

Author(s):

Definition   A set of m positive integers $ \{a_1, a_2, \dots, a_m\} $ is called a Diophantine $ m $-tuple if $ a_i\cdot a_j + 1 $ is a perfect square for all $ 1 \leq i < j \leq m $.
Conjecture  (1)   Diophantine quintuple does not exist.

It would follow from the following stronger conjecture [Da]:

Conjecture  (2)   If $ \{a, b, c, d\} $ is a Diophantine quadruple and $ d > \max \{a, b, c\} $, then $ d = a + b + c + 2bc + 2\sqrt{(ab+1)(ac+1)(bc+1)}. $

Keywords:

Drawing disconnected graphs on surfaces ★★

Author(s): DeVos; Mohar; Samal

Conjecture   Let $ G $ be the disjoint union of the graphs $ G_1 $ and $ G_2 $ and let $ \Sigma $ be a surface. Is it true that every optimal drawing of $ G $ on $ \Sigma $ has the property that $ G_1 $ and $ G_2 $ are disjoint?

Keywords: crossing number; surface