Random

Choice Number of k-Chromatic Graphs of Bounded Order ★★

Author(s): Noel

Conjecture   If $ G $ is a $ k $-chromatic graph on at most $ mk $ vertices, then $ \text{ch}(G)\leq \text{ch}(K_{m*k}) $.

Keywords: choosability; complete multipartite graph; list coloring

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

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Free Hollywood Story Free Diamonds Gems Cheats 2024 (Safe)

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Genshin Impact Cheats Generator 2024 Edition Update (WORKS) ★★

Author(s):

Genshin Impact Cheats Generator 2024 Edition Update (WORKS)

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Free Geometry Dash Cheats Gold Coins Stars Generator 2023-2024 ★★

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Free Geometry Dash Cheats Gold Coins Stars Generator 2023-2024

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Rendezvous on a line ★★

Author(s):

Rendezvous on a line

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Cyclic spanning subdigraph with small cyclomatic number ★★

Author(s): Bondy

Conjecture   Let $ D $ be a digraph all of whose strong components are nontrivial. Then $ D $ contains a cyclic spanning subdigraph with cyclomatic number at most $ \alpha(D) $.

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Critical Ops Unlimited Credits Cheats IOS Android No Survey 2024 (FREE METHOD) ★★

Author(s):

Critical Ops Unlimited Credits Cheats IOS Android No Survey 2024 (FREE METHOD)

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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

Every metamonovalued funcoid is monovalued ★★

Author(s): Porton

Conjecture   Every metamonovalued funcoid is monovalued.

The reverse is almost trivial: Every monovalued funcoid is metamonovalued.

Keywords: monovalued

Idle Miner Tycoon Cheats Generator Pro Apk (Android Ios) ★★

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Idle Miner Tycoon Cheats Generator Pro Apk (Android Ios)

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MacEachen Conjecture

Author(s): McEachen

Conjecture   Every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product.

Keywords: primality; prime distribution

Which homology 3-spheres bound homology 4-balls? ★★★★

Author(s): Ancient/folklore

Problem   Is there a complete and computable set of invariants that can determine which (rational) homology $ 3 $-spheres bound (rational) homology $ 4 $-balls?

Keywords: cobordism; homology ball; homology sphere

Hall-Paige conjecture (Solved) ★★

Author(s):

Hall-Paige conjecture (Solved)

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Circular flow numbers of $r$-graphs ★★

Author(s): Steffen

A nowhere-zero $ r $-flow $ (D(G),\phi) $ on $ G $ is an orientation $ D $ of $ G $ together with a function $ \phi $ from the edge set of $ G $ into the real numbers such that $ 1 \leq |\phi(e)| \leq r-1 $, for all $ e \in E(G) $, and $ \sum_{e \in E^+(v)}\phi(e) = \sum_{e \in E^-(v)}\phi(e), \textrm{ for all } v \in V(G) $.

A $ (2t+1) $-regular graph $ G $ is a $ (2t+1) $-graph if $ |\partial_G(X)| \geq 2t+1 $ for every $ X \subseteq V(G) $ with $ |X| $ odd.

Conjecture   Let $ t > 1 $ be an integer. If $ G $ is a $ (2t+1) $-graph, then $ F_c(G) \leq 2 + \frac{2}{t} $.

Keywords: flow conjectures; nowhere-zero flows

Easy! Unlimited Rise Of Kingdoms Cheats Generator codes (GLITCH) ★★

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Easy! Unlimited Rise Of Kingdoms Cheats Generator codes (GLITCH)

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Rainbow Six Siege Cheats Generator Unlimited R6 No Jailbreak (Premium Orginal Generator) ★★

Author(s):

Rainbow Six Siege Cheats Generator Unlimited R6 No Jailbreak (Premium Orginal Generator)

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Criterion for boundedness of power series

Author(s): Rüdinger

Question   Give a necessary and sufficient criterion for the sequence $ (a_n) $ so that the power series $ \sum_{n=0}^{\infty} a_n x^n $ is bounded for all $ x \in \mathbb{R} $.

Keywords: boundedness; power series; real analysis

Matchington Mansion Stars Coins Cheats IOS And Android No Verification Generator 2024 (fresh method) ★★

Author(s):

Matchington Mansion Stars Coins Cheats IOS And Android No Verification Generator 2024 (fresh method)

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A conjecture about direct product of funcoids ★★

Author(s): Porton

Conjecture   Let $ f_1 $ and $ f_2 $ are monovalued, entirely defined funcoids with $ \operatorname{Src}f_1=\operatorname{Src}f_2=A $. Then there exists a pointfree funcoid $ f_1 \times^{\left( D \right)} f_2 $ such that (for every filter $ x $ on $ A $) $$\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X \times^{\mathsf{FCD}} \langle f_2\rangle X \hspace{1em} | \hspace{1em} X \in \mathrm{atoms}^{\mathfrak{A}} x \right\}.$$ (The join operation is taken on the lattice of filters with reversed order.)

A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.

Keywords: category theory; general topology

Covering systems with big moduli ★★

Author(s): Erdos; Selfridge

Problem   Does for every integer $ N $ exist a covering system with all moduli distinct and at least equal to~$ N $?

Keywords: covering system

Bingo Blitz Cheats Generator iOS Android (Current 2024 Generator) ★★

Author(s):

Bingo Blitz Cheats Generator iOS Android (Current 2024 Generator)

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P vs. NP ★★★★

Author(s): Cook; Levin

Problem   Is P = NP?

Keywords: Complexity Class; Computational Complexity; Millenium Problems; NP; P; polynomial algorithm

Twin prime conjecture ★★★★

Author(s):

Conjecture   There exist infinitely many positive integers $ n $ so that both $ n $ and $ n+2 $ are prime.

Keywords: prime; twin prime

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

Growth of finitely presented groups ★★★

Author(s): Adyan

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

Keywords: finitely presented; growth

Singmaster's conjecture ★★

Author(s): Singmaster

Conjecture   There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number $ 1 $.

The number $ 2 $ appears once in Pascal's triangle, $ 3 $ appears twice, $ 6 $ appears three times, and $ 10 $ appears $ 4 $ times. There are infinite families of numbers known to appear $ 6 $ times. The only number known to appear $ 8 $ times is $ 3003 $. It is not known whether any number appears more than $ 8 $ times. The conjectured upper bound could be $ 8 $; Singmaster thought it might be $ 10 $ or $ 12 $. See Singmaster's conjecture.

Keywords: Pascal's triangle

War Thunder Golden Eagles Cheats IOS And Android No Verification Generator 2024 (fresh method) ★★

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War Thunder Golden Eagles Cheats IOS And Android No Verification Generator 2024 (fresh method)

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Easy! Unlimited Dragon City Cheats Generator codes (GLITCH) ★★

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Easy! Unlimited Dragon City Cheats Generator codes (GLITCH)

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Free DealDash Bids Cheats Bids Generator 2023-2024 ★★

Author(s):

Free DealDash Bids Cheats Bids Generator 2023-2024

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Partition of Complete Geometric Graph into Plane Trees ★★

Author(s):

Conjecture   Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.

Keywords: complete geometric graph, edge colouring

Monochromatic reachability in arc-colored digraphs ★★★

Author(s): Sands; Sauer; Woodrow

Conjecture   For every $ k $, there exists an integer $ f(k) $ such that if $ D $ is a digraph whose arcs are colored with $ k $ colors, then $ D $ has a $ S $ set which is the union of $ f(k) $ stables sets so that every vertex has a monochromatic path to some vertex in $ S $.

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r-regular graphs are not uniquely hamiltonian. ★★★

Author(s): Sheehan

Conjecture   If $ G $ is a finite $ r $-regular graph, where $ r > 2 $, then $ G $ is not uniquely hamiltonian.

Keywords: hamiltonian; regular; uniquely hamiltonian

Fishdom Cheats Generator without verification (Free) ★★

Author(s):

Fishdom Cheats Generator without verification (Free)

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New-mathod! Free Bloons TD Battles Energy Medal Money Cheats 2024 (No Human Verification) ★★

Author(s):

New-mathod! Free Bloons TD Battles Energy Medal Money Cheats 2024 (No Human Verification)

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F_d versus F_{d+1} ★★★

Author(s): Krajicek

Problem   Find a constant $ k $ such that for any $ d $ there is a sequence of tautologies of depth $ k $ that have polynomial (or quasi-polynomial) size proofs in depth $ d+1 $ Frege system $ F_{d+1} $ but requires exponential size $ F_d $ proofs.

Keywords: Frege system; short proof

Edge-Colouring Geometric Complete Graphs ★★

Author(s): Hurtado

Question   What is the minimum number of colours such that every complete geometric graph on $ n $ vertices has an edge colouring such that:
    \item[Variant A] crossing edges get distinct colours, \item[Variant B] disjoint edges get distinct colours, \item[Variant C] non-disjoint edges get distinct colours, \item[Variant D] non-crossing edges get distinct colours.

Keywords: geometric complete graph, colouring

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

Author(s):

Match Masters Free Coins Cheats 2024 (FREE!)

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Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems) ★★★★

Author(s):

Conjecture   Let $ Diff^{r}(M)  $ be the space of $ C^{r} $ Diffeomorphisms on the connected , compact and boundaryles manifold M and $ \chi^{r}(M) $ the space of $ C^{r} $ vector fields. There is a dense set $ D\subset Diff^{r}(M) $ ($ D\subset \chi^{r}(M) $ ) such that $ \forall f\in D $ exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space $ M $

This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .

Keywords: Attractors , basins, Finite

Chromatic number of associahedron ★★

Author(s): Fabila-Monroy; Flores-Penaloza; Huemer; Hurtado; Urrutia; Wood

Conjecture   Associahedra have unbounded chromatic number.

Keywords: associahedron, graph colouring, chromatic number

Every 4-connected toroidal graph has a Hamilton cycle ★★

Author(s): Grunbaum; Nash-Williams

Conjecture   Every 4-connected toroidal graph has a Hamilton cycle.

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Shuffle-Exchange Conjecture (graph-theoretic form) ★★★

Author(s): Beneš; Folklore; Stone

Given integers $ k,n \ge 2 $, the 2-stage Shuffle-Exchange graph/network, denoted $ \text{SE}(k,n) $, is the simple $ k $-regular bipartite graph with the ordered pair $ (U,V) $ of linearly labeled parts $ U:=\{u_0,\dots,u_{t-1}\} $ and $ V:=\{v_0,\dots,v_{t-1}\} $, where $ t:=k^{n-1} $, such that vertices $ u_i $ and $ v_j $ are adjacent if and only if $ (j - ki) \text{ mod } t < k $ (see Fig.1).

Given integers $ k,n,r \ge 2 $, the $ r $-stage Shuffle-Exchange graph/network, denoted $ (\text{SE}(k,n))^{r-1} $, is the proper (i.e., respecting all the orders) concatenation of $ r-1 $ identical copies of $ \text{SE}(k,n) $ (see Fig.1).

Let $ r(k,n) $ be the smallest integer $ r\ge 2 $ such that the graph $ (\text{SE}(k,n))^{r-1} $ is rearrangeable.

Problem   Find $ r(k,n) $.
Conjecture   $ r(k,n)=2n-1 $.

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Bleach Brave Souls Cheats Generator No Human Verification (Without Surveys) ★★

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Bleach Brave Souls Cheats Generator No Human Verification (Without Surveys)

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Apex Legends Coins Cheats 2024 (rejuvenated cheats) ★★

Author(s):

Apex Legends Coins Cheats 2024 (rejuvenated cheats)

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Every metamonovalued reloid is monovalued ★★

Author(s): Porton

Conjecture   Every metamonovalued reloid is monovalued.

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Dragon Ball Z Dokkan Battle Cheats Generator 2024 (LEGIT) ★★

Author(s):

Dragon Ball Z Dokkan Battle Cheats Generator 2024 (LEGIT)

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Hello ★★

Author(s):

Hello

http://www.openproblemgarden.org/op/hello

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FarmVille 2 Cheats Coins Farm Bucks Generator Tested on iOS and Android (Latest Method) ★★

Author(s):

FarmVille 2 Cheats Coins Farm Bucks Generator Tested on iOS and Android (Latest Method)

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Decomposing k-arc-strong tournament into k spanning strong digraphs ★★

Author(s): Bang-Jensen; Yeo

Conjecture   Every k-arc-strong tournament decomposes into k spanning strong digraphs.

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Monochromatic vertex colorings inherited from Perfect Matchings ★★★

Author(s):

Conjecture   For which values of $ n $ and $ d $ are there bi-colored graphs on $ n $ vertices and $ d $ different colors with the property that all the $ d $ monochromatic colorings have unit weight, and every other coloring cancels out?

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MONOPOLY GO Cheats Generator IOS Android No Verification 2024 (fresh method) ★★

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MONOPOLY GO Cheats Generator IOS Android No Verification 2024 (fresh method)

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