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Unfriendly partitions ★★★

Author(s): Cowan; Emerson

If $ G $ is a graph, we say that a partition of $ V(G) $ is unfriendly if every vertex has at least as many neighbors in the other classes as in its own.

Problem   Does every countably infinite graph have an unfriendly partition into two sets?

Keywords: coloring; infinite graph; partition

Free Clash of Clans Gems Cheats 2024 Edition Update (WORKS!) ★★

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Free Clash of Clans Gems Cheats 2024 Edition Update (WORKS!)

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Unused Free Bloons TD Battles Cheats No Human Verification No Survey (2024 Method) ★★

Author(s):

Unused Free Bloons TD Battles Cheats No Human Verification No Survey (2024 Method)

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Colouring the square of a planar graph ★★

Author(s): Wegner

Conjecture   Let $ G $ be a planar graph of maximum degree $ \Delta $. The chromatic number of its square is
    \item at most $ 7 $ if $ \Delta =3 $, \item at most $ \Delta+5 $ if $ 4\leq\Delta\leq 7 $, \item at most $ \left\lfloor\frac32\,\Delta\right\rfloor+1 $ if $ \Delta\ge8 $.

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Linial-Berge path partition duality ★★★

Author(s): Berge; Linial

Conjecture   The minimum $ k $-norm of a path partition on a directed graph $ D $ is no more than the maximal size of an induced $ k $-colorable subgraph.

Keywords: coloring; directed path; partition

Hungry Shark Evolution Cheats Generator IOS Android No Survey 2024 (Generator!) ★★

Author(s):

Hungry Shark Evolution Cheats Generator IOS Android No Survey 2024 (Generator!)

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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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Jorgensen's Conjecture ★★★

Author(s): Jorgensen

Conjecture   Every 6-connected graph without a $ K_6 $ minor is apex (planar plus one vertex).

Keywords: connectivity; minor

Codes Free Star Stable Star Coins Jorvik Coins Cheats 2024 No Human Veryfication!!! ★★

Author(s):

Codes Free Star Stable Star Coins Jorvik Coins Cheats 2024 No Human Veryfication!!!

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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

Legal* Free Coin Master Cheats Spins Coins Generator No Human Verification 2024 ★★

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Legal* Free Coin Master Cheats Spins Coins Generator No Human Verification 2024

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Crossing numbers and coloring ★★★

Author(s): Albertson

We let $ cr(G) $ denote the crossing number of a graph $ G $.

Conjecture   Every graph $ G $ with $ \chi(G) \ge t $ satisfies $ cr(G) \ge cr(K_t) $.

Keywords: coloring; complete graph; crossing number

Toon Blast Cheats Generator 2024 Cheats Generator Tested On Android Ios (extra) ★★

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Toon Blast Cheats Generator 2024 Cheats Generator Tested On Android Ios (extra)

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Fortnite Working Generator V-Bucks Generator (NEW AND FREE) ★★

Author(s):

Fortnite Working Generator V-Bucks Generator (NEW AND FREE)

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Dragon City Cheats Generator 2024 Update Hacks (Verified) ★★

Author(s):

Dragon City Cheats Generator 2024 Update Hacks (Verified)

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Saturated $k$-Sperner Systems of Minimum Size ★★

Author(s): Morrison; Noel; Scott

Question   Does there exist a constant $ c>1/2 $ and a function $ n_0(k) $ such that if $ |X|\geq n_0(k) $, then every saturated $ k $-Sperner system $ \mathcal{F}\subseteq \mathcal{P}(X) $ has cardinality at least $ 2^{(1+o(1))ck} $?

Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system

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Author(s):

"New Cheats" Subway Surfers Coins Keys Cheats Free 2024

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Hamiltonian paths and cycles in vertex transitive graphs ★★★

Author(s): Lovasz

Problem   Does every connected vertex-transitive graph have a Hamiltonian path?

Keywords: cycle; hamiltonian; path; vertex-transitive

Hungry Shark World Cheats Generator 2024 (fresh strategy) ★★

Author(s):

Hungry Shark World Cheats Generator 2024 (fresh strategy)

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Decomposing an eulerian graph into cycles. ★★

Author(s): Hajós

Conjecture   Every simple eulerian graph on $ n $ vertices can be decomposed into at most $ \frac{1}{2}(n-1) $ cycles.

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Cores of strongly regular graphs ★★★

Author(s): Cameron; Kazanidis

Question   Does every strongly regular graph have either itself or a complete graph as a core?

Keywords: core; strongly regular

Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament ★★

Author(s): Yuster

Conjecture   If $ T $ is a tournament of order $ n $, then it contains $ \left \lceil n(n-1)/6 - n/3\right\rceil $ arc-disjoint transitive subtournaments of order 3.

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Clash of Clans Gems Cheats without verification (Free) ★★

Author(s):

Clash of Clans Gems Cheats without verification (Free)

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Golf Battle Cheats Generator (Ios Android) ★★

Author(s):

Golf Battle Cheats Generator (Ios Android)

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Friendly partitions ★★

Author(s): DeVos

A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.

Problem   Is it true that for every $ r $, all but finitely many $ r $-regular graphs have friendly partitions?

Keywords: edge-cut; partition; regular

Simpsons Tapped Out Cheats Generator (New Working Cheats Generator 2024) ★★

Author(s):

Simpsons Tapped Out Cheats Generator (New Working Cheats Generator 2024)

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My Singing Monsters Cheats Generator 2024 Cheats Generator Tested On Android Ios (extra) ★★

Author(s):

My Singing Monsters Cheats Generator 2024 Cheats Generator Tested On Android Ios (extra)

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Growth of finitely presented groups ★★★

Author(s): Adyan

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

Keywords: finitely presented; growth

Extremal problem on the number of tree endomorphism ★★

Author(s): Zhicong Lin

Conjecture   An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the $ n $ vertices' trees, the star with $ n $ vertices has the most endomorphisms, while the path with $ n $ vertices has the least endomorphisms.

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Switching reconstruction of digraphs ★★

Author(s): Bondy; Mercier

Question   Are there any switching-nonreconstructible digraphs on twelve or more vertices?

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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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Alexa's Conjecture on Primality ★★

Author(s): Alexa

Definition   Let $ r_i $ be the unique integer (with respect to a fixed $ p\in\mathbb{N} $) such that

$$(2i+1)^{p-1} \equiv r_i \pmod p ~~\text{ and } ~ 0 \le r_i < p. $$

Conjecture   A natural number $ p \ge 8 $ is a prime iff $$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$

Keywords: primality

Genshin Impact Cheats Generator 2023-2024 Edition Hack (NEW-FREE!!) ★★

Author(s):

Genshin Impact Cheats Generator 2023-2024 Edition Hack (NEW-FREE!!)

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Mapping planar graphs to odd cycles ★★★

Author(s): Jaeger

Conjecture   Every planar graph of girth $ \ge 4k $ has a homomorphism to $ C_{2k+1} $.

Keywords: girth; homomorphism; planar graph

Brawlhalla Cheats Generator 2024 Real Working (new method) ★★

Author(s):

Brawlhalla Cheats Generator 2024 Real Working (new method)

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Dividing up the unrestricted partitions ★★

Author(s): David S.; Newman

Begin with the generating function for unrestricted partitions:

(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...

Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.

Keywords: congruence properties; partition

Legal* Free Warzone Cheats COD points Generator No Human Verification 2024 ★★

Author(s):

Legal* Free Warzone Cheats COD points Generator No Human Verification 2024

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Cheats Free* Warzone COD points Cheats 2024 No Human Verification ★★

Author(s):

Cheats Free* Warzone COD points Cheats 2024 No Human Verification

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Circular colouring the orthogonality graph ★★

Author(s): DeVos; Ghebleh; Goddyn; Mohar; Naserasr

Let $ {\mathcal O} $ denote the graph with vertex set consisting of all lines through the origin in $ {\mathbb R}^3 $ and two vertices adjacent in $ {\mathcal O} $ if they are perpendicular.

Problem   Is $ \chi_c({\mathcal O}) = 4 $?

Keywords: circular coloring; geometric graph; orthogonality

Approximation ratio for k-outerplanar graphs ★★

Author(s): Bentz

Conjecture   Is the approximation ratio for the Maximum Edge Disjoint Paths (MaxEDP) or the Maximum Integer Multiflow problem (MaxIMF) bounded by a constant in $ k $-outerplanar graphs or tree-width graphs?

Keywords: approximation algorithms; planar graph; polynomial algorithm

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

Working Apex Legends Cheats Online Coins Generator (No Survey) ★★

Author(s):

Working Apex Legends Cheats Online Coins Generator (No Survey)

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3-flow conjecture ★★★

Author(s): Tutte

Conjecture   Every 4-edge-connected graph has a nowhere-zero 3-flow.

Keywords: nowhere-zero flow

House Of Fun Cheats Generator Free Cheats Generator 2024 No Verification (Android iOS) ★★

Author(s):

House Of Fun Cheats Generator Free Cheats Generator 2024 No Verification (Android iOS)

Keywords:

Geometry Dash Gold Coins Stars Cheats 2024 Update (FREE!!) ★★

Author(s):

Geometry Dash Gold Coins Stars Cheats 2024 Update (FREE!!)

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Lords Mobile Working Cheats Gems Coins Generator (NEW AND FREE) ★★

Author(s):

Lords Mobile Working Cheats Gems Coins Generator (NEW AND FREE)

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Open problem ★★

Author(s):

Open problem

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Average diameter of a bounded cell of a simple arrangement ★★

Author(s): Deza; Terlaky; Zinchenko

Conjecture   The average diameter of a bounded cell of a simple arrangement defined by $ n $ hyperplanes in dimension $ d $ is not greater than $ d $.

Keywords: arrangement; diameter; polytope

List Colourings of Complete Multipartite Graphs with 2 Big Parts ★★

Author(s): Allagan

Question   Given $ a,b\geq2 $, what is the smallest integer $ t\geq0 $ such that $ \chi_\ell(K_{a,b}+K_t)= \chi(K_{a,b}+K_t) $?

Keywords: complete bipartite graph; complete multipartite graph; list coloring

Are there only finite Fermat Primes? ★★★

Author(s):

Conjecture   A Fermat prime is a Fermat number \[ F_n  = 2^{2^n }  + 1 \] that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.

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