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Hamilton decomposition of prisms over 3-connected cubic planar graphs ★★

Author(s): Alspach; Rosenfeld

Conjecture   Every prism over a $ 3 $-connected cubic planar graph can be decomposed into two Hamilton cycles.

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

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

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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

Raid Shadow Legends Cheats Generator 2024 (fresh strategy) ★★

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Raid Shadow Legends Cheats Generator 2024 (fresh strategy)

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The Crossing Number of the Complete Bipartite Graph ★★★

Author(s): Turan

The crossing number $ cr(G) $ of $ G $ is the minimum number of crossings in all drawings of $ G $ in the plane.

Conjecture   $ \displaystyle   cr(K_{m,n}) = \floor{\frac m2} \floor{\frac {m-1}2}                      \floor{\frac n2} \floor{\frac {n-1}2}  $

Keywords: complete bipartite graph; crossing number

Erdős-Posa property for long directed cycles ★★

Author(s): Havet; Maia

Conjecture   Let $ \ell \geq 2 $ be an integer. For every integer $ n\geq 0 $, there exists an integer $ t_n=t_n(\ell) $ such that for every digraph $ D $, either $ D $ has a $ n $ pairwise-disjoint directed cycles of length at least $ \ell $, or there exists a set $ T $ of at most $ t_n $ vertices such that $ D-T $ has no directed cycles of length at least $ \ell $.

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Free Coin Master Cheats No Human Verification No Survey (2024 Method) ★★

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Free Coin Master Cheats No Human Verification No Survey (2024 Method)

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eFootball 2023 Cheats Generator Unlimited IOS Android No Survey 2024 (FREE METHOD) ★★

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eFootball 2023 Cheats Generator Unlimited IOS Android No Survey 2024 (FREE METHOD)

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Warframe Cheats Generator (iOS Android 2024) ★★

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Warframe Cheats Generator (iOS Android 2024)

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Olson's Conjecture ★★

Author(s): Olson

Conjecture   If $ a_1,a_2,\ldots,a_{2n-1} $ is a sequence of elements from a multiplicative group of order $ n $, then there exist $ 1 \le j_1 < j_2 \ldots < j_n \le 2n-1 $ so that $ \prod_{i=1}^n a_{j_i} = 1 $.

Keywords: zero sum

Match Masters Free Coins Cheats 2024 (LEGIT) ★★

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Match Masters Free Coins Cheats 2024 (LEGIT)

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Bingo Blitz Cheats Generator Unlimited No Jailbreak (Premium) ★★

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Bingo Blitz Cheats Generator Unlimited No Jailbreak (Premium)

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Marvel Strike Force Cheats Generator Android Ios 2024 Cheats Generator (improved version) ★★

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Marvel Strike Force Cheats Generator Android Ios 2024 Cheats Generator (improved version)

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Outward reloid of composition vs composition of outward reloids ★★

Author(s): Porton

Conjecture   For every composable funcoids $ f $ and $ g $ $$(\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq(\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{RLD})_{\mathrm{out}}f.$$

Keywords: outward reloid

Upgrading a completary multifuncoid ★★

Author(s): Porton

Let $ \mho $ be a set, $ \mathfrak{F} $ be the set of filters on $ \mho $ ordered reverse to set-theoretic inclusion, $ \mathfrak{P} $ be the set of principal filters on $ \mho $, let $ n $ be an index set. Consider the filtrator $ \left( \mathfrak{F}^n ; \mathfrak{P}^n \right) $.

Conjecture   If $ f $ is a completary multifuncoid of the form $ \mathfrak{P}^n $, then $ E^{\ast} f $ is a completary multifuncoid of the form $ \mathfrak{F}^n $.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

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Highly arc transitive two ended digraphs ★★

Author(s): Cameron; Praeger; Wormald

Conjecture   If $ G $ is a highly arc transitive digraph with two ends, then every tile of $ G $ is a disjoint union of complete bipartite graphs.

Keywords: arc transitive; digraph; infinite graph

Bleach Brave Souls Cheats Generator Free 2024 No Human Verification (New Update) ★★

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Bleach Brave Souls Cheats Generator Free 2024 No Human Verification (New Update)

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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

Convex 'Fair' Partitions Of Convex Polygons ★★

Author(s): Nandakumar; Ramana

Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and same perimeter?

Definitions: Define a Fair Partition of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a Convex Fair Partition.

Questions: 1. (Rephrasing the above 'basic' question) Given any positive integer n, can any convex polygon be convex fair partitioned into n pieces?

2. If the answer to the above is "Not always'', how does one decide the possibility of such a partition for a given convex polygon and a given n? And if fair convex partition is allowed by a specific convex polygon for a give n, how does one find the optimal convex fair partition that minimizes the total length of the cut segments?

3. Finally, what could one say about higher dimensional analogs of this question?

Conjecture: The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: Every convex polygon allows a convex fair partition into n pieces for any n

Keywords: Convex Polygons; Partitioning

Characterizing (aleph_0,aleph_1)-graphs ★★★

Author(s): Diestel; Leader

Call a graph an $ (\aleph_0,\aleph_1) $-graph if it has a bipartition $ (A,B) $ so that every vertex in $ A $ has degree $ \aleph_0 $ and every vertex in $ B $ has degree $ \aleph_1 $.

Problem   Characterize the $ (\aleph_0,\aleph_1) $-graphs.

Keywords: binary tree; infinite graph; normal spanning tree; set theory

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.

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War Machines Coins Diamonds Cheats 2024 (iOS Android) ★★

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Conjecture  

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Stable set meeting all longest directed paths. ★★

Author(s): Laborde; Payan; Xuong N.H.

Conjecture   Every digraph has a stable set meeting all longest directed paths

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Cookie Run Kingdom Cheats Generator Unlimited Cheats Generator IOS Android 2024 (get codes) ★★

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Cookie Run Kingdom Cheats Generator Unlimited Cheats Generator IOS Android 2024 (get codes)

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Free Call Of Duty Mobile Cheats Generator No Human Verification No Survey (Unused) ★★

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Free Call Of Duty Mobile Cheats Generator No Human Verification No Survey (Unused)

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Free Bloons TD Battles Energy Medal Money Cheats Pro Apk 2024 (Android Ios) ★★

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Free Bloons TD Battles Energy Medal Money Cheats Pro Apk 2024 (Android Ios)

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V-Bucks Generator Free 2024 in 5 minutes (New Generator V-Bucks) ★★

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V-Bucks Generator Free 2024 in 5 minutes (New Generator V-Bucks)

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Legal SimCity BuildIt Cheats Generator No Human Verification 2024 (No Surveys Needed) ★★

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Legal SimCity BuildIt Cheats Generator No Human Verification 2024 (No Surveys Needed)

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Fractional Hadwiger ★★

Author(s): Harvey; Reed; Seymour; Wood

Conjecture   For every graph $ G $,
(a) $ \chi_f(G)\leq\text{had}(G) $
(b) $ \chi(G)\leq\text{had}_f(G) $
(c) $ \chi_f(G)\leq\text{had}_f(G) $.

Keywords: fractional coloring, minors

House Of Fun Cheats Generator Free Unlimited Cheats Generator (new codes Generator) ★★

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House Of Fun Cheats Generator Free Unlimited Cheats Generator (new codes Generator)

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Subgraph of large average degree and large girth. ★★

Author(s): Thomassen

Conjecture   For all positive integers $ g $ and $ k $, there exists an integer $ d $ such that every graph of average degree at least $ d $ contains a subgraph of average degree at least $ k $ and girth greater than $ g $.

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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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Multicolour Erdős--Hajnal Conjecture ★★★

Author(s): Erdos; Hajnal

Conjecture   For every fixed $ k\geq2 $ and fixed colouring $ \chi $ of $ E(K_k) $ with $ m $ colours, there exists $ \varepsilon>0 $ such that every colouring of the edges of $ K_n $ contains either $ k $ vertices whose edges are coloured according to $ \chi $ or $ n^\varepsilon $ vertices whose edges are coloured with at most $ m-1 $ colours.

Keywords: ramsey theory

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

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

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Blatter-Specker Theorem for ternary relations ★★

Author(s): Makowsky

Let $ C $ be a class of finite relational structures. We denote by $ f_C(n) $ the number of structures in $ C $ over the labeled set $ \{0, \dots, n-1 \} $. For any class $ C $ definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every $ m \in \mathbb{N} $, the function $ f_C(n) $ is ultimately periodic modulo $ m $.

Question   Does the Blatter-Specker Theorem hold for ternary relations.

Keywords: Blatter-Specker Theorem; FMT00-Luminy

Warframe Free Platinum Cheats Free Generator 2024 in 5 minutes (successive cheats) ★★

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Warframe Free Platinum Cheats Free Generator 2024 in 5 minutes (successive cheats)

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World Of Tanks Blitz Gold Credits Cheats Generator 2024 (improved version) ★★

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World Of Tanks Blitz Gold Credits Cheats Generator 2024 (improved version)

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Primitive pythagorean n-tuple tree ★★

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Conjecture   Find linear transformation construction of primitive pythagorean n-tuple tree!

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Boom Beach Unlimited Diamonds Cheats Generator 2024 (fresh strategy) ★★

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Boom Beach Unlimited Diamonds Cheats Generator 2024 (fresh strategy)

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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

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

(m,n)-cycle covers ★★★

Author(s): Celmins; Preissmann

Conjecture   Every bridgeless graph has a (5,2)-cycle-cover.

Keywords: cover; cycle

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

Faithful cycle covers ★★★

Author(s): Seymour

Conjecture   If $ G = (V,E) $ is a graph, $ p : E \rightarrow {\mathbb Z} $ is admissable, and $ p(e) $ is even for every $ e \in E(G) $, then $ (G,p) $ has a faithful cover.

Keywords: cover; cycle

Reed's omega, delta, and chi conjecture ★★★

Author(s): Reed

For a graph $ G $, we define $ \Delta(G) $ to be the maximum degree, $ \omega(G) $ to be the size of the largest clique subgraph, and $ \chi(G) $ to be the chromatic number of $ G $.

Conjecture   $ \chi(G) \le \ceil{\frac{1}{2}(\Delta(G)+1) + \frac{1}{2}\omega(G)} $ for every graph $ G $.

Keywords: coloring

Forcing a $K_6$-minor ★★

Author(s): Barát ; Joret; Wood

Conjecture   Every graph with minimum degree at least 7 contains a $ K_6 $-minor.
Conjecture   Every 7-connected graph contains a $ K_6 $-minor.

Keywords: connectivity; graph minors

The Hodge Conjecture ★★★★

Author(s): Hodge

Conjecture   Let $ X $ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of $ X $.

Keywords: Hodge Theory; Millenium Problems

Good Edge Labelings ★★

Author(s): Araújo; Cohen; Giroire; Havet

Question   What is the maximum edge density of a graph which has a good edge labeling?

We say that a graph is good-edge-labeling critical, if it has no good edge labeling, but every proper subgraph has a good edge labeling.

Conjecture   For every $ c<4 $, there is only a finite number of good-edge-labeling critical graphs with average degree less than $ c $.

Keywords: good edge labeling, edge labeling

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

Inequality of the means ★★★

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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