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Counterexamples to the Baillie-PSW primality test ★★

Author(s):

Problem  (1)   Find a counterexample to Baillie-PSW primality test or prove that there is no one.
Problem  (2)   Find a composite $ n\equiv 3 $ or $ 7\pmod{10} $ which divides both $ 2^{n-1} - 1 $ (see Fermat pseudoprime) and the Fibonacci number $ F_{n+1} $ (see Lucas pseudoprime), or prove that there is no such $ n $.

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Counting 3-colorings of the hex lattice ★★

Author(s): Thomassen

Problem   Find $ \lim_{n \rightarrow \infty} (\chi( H_n , 3)) ^{ 1 / |V(H_n)| } $.

Keywords: coloring; Lieb's Ice Constant; tiling; torus

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

Author(s):

My Singing Monsters Cheats Generator 2024 (rejuvenated Generator)

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Star chromatic index of cubic graphs ★★

Author(s): Dvorak; Mohar; Samal

The star chromatic index $ \chi_s'(G) $ of a graph $ G $ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.

Question   Is it true that for every (sub)cubic graph $ G $, we have $ \chi_s'(G) \le 6 $?

Keywords: edge coloring; star coloring

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

Author(s):

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

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Monotone 4-term Arithmetic Progressions ★★

Author(s): Davis; Entringer; Graham; Simmons

Question   Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions?

Keywords: monotone arithmetic progression; permutation

Cores of Cayley graphs ★★

Author(s): Samal

Conjecture   Let $ M $ be an abelian group. Is the core of a Cayley graph (on some power of $ M $) a Cayley graph (on some power of $ M $)?

Keywords: Cayley graph; core

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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Kriesell's Conjecture ★★

Author(s): Kriesell

Conjecture   Let $ G $ be a graph and let $ T\subseteq V(G) $ such that for any pair $ u,v\in T $ there are $ 2k $ edge-disjoint paths from $ u $ to $ v $ in $ G $. Then $ G $ contains $ k $ edge-disjoint trees, each of which contains $ T $.

Keywords: Disjoint paths; edge-connectivity; spanning trees

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 $.

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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

trace inequality ★★

Author(s):

Let $ A,B $ be positive semidefinite, by Jensen's inequality, it is easy to see $ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $, whenever $ s>r>0 $.

What about the $ tr(A^s+B^s)^{\frac{1}{s}}\leq tr(A^r+B^r)^{\frac{1}{r}} $, is it still valid?

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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

New-mathod! Free Kim Kardashian Hollywood Cash Stars Cheats 2024 (No Human Verification) ★★

Author(s):

New-mathod! Free Kim Kardashian Hollywood Cash Stars Cheats 2024 (No Human Verification)

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A sextic counterexample to Euler's sum of powers conjecture ★★

Author(s): Euler

Problem   Find six positive integers $ x_1, x_2, \dots, x_6 $ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do not exist.

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Signing a graph to have small magnitude eigenvalues ★★

Author(s): Bilu; Linial

Conjecture   If $ A $ is the adjacency matrix of a $ d $-regular graph, then there is a symmetric signing of $ A $ (i.e. replace some $ +1 $ entries by $ -1 $) so that the resulting matrix has all eigenvalues of magnitude at most $ 2 \sqrt{d-1} $.

Keywords: eigenvalue; expander; Ramanujan graph; signed graph; signing

Seymour's Second Neighbourhood Conjecture ★★★

Author(s): Seymour

Conjecture   Any oriented graph has a vertex whose outdegree is at most its second outdegree.

Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour

Graham's conjecture on tree reconstruction ★★

Author(s): Graham

Problem   for every graph $ G $, we let $ L(G) $ denote the line graph of $ G $. Given that $ G $ is a tree, can we determine it from the integer sequence $ |V(G)|, |V(L(G))|, |V(L(L(G)))|, \ldots $?

Keywords: reconstruction; tree

Hall-Paige conjecture (Solved) ★★

Author(s):

Hall-Paige conjecture (Solved)

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

Author(s):

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

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Real Racing 3 Cheats Generator Working 2024 (Real Racing 3 Generator) ★★

Author(s):

Real Racing 3 Cheats Generator Working 2024 (Real Racing 3 Generator)

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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

Critical Ops Unlimited Credits Cheats IOS Android No Survey 2024 (Reedem Today) ★★

Author(s):

Critical Ops Unlimited Credits Cheats IOS Android No Survey 2024 (Reedem Today)

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Idle Miner Tycoon Cheats Generator 2023-2024 (No Human Verification) ★★

Author(s):

Idle Miner Tycoon Cheats Generator 2023-2024 (No Human Verification)

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Turán Problem for $10$-Cycles in the Hypercube ★★

Author(s): Erdos

Problem   Bound the extremal number of $ C_{10} $ in the hypercube.

Keywords: cycles; extremal combinatorics; hypercube

2-accessibility of primes ★★

Author(s): Landman; Robertson

Question   Is the set of prime numbers 2-accessible?

Keywords: monochromatic diffsequences; primes

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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Splitting a digraph with minimum outdegree constraints ★★★

Author(s): Alon

Problem   Is there a minimum integer $ f(d) $ such that the vertices of any digraph with minimum outdegree $ d $ can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least $ d $?

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Circular flow number of regular class 1 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) $. The circular flow number of $ G $ is inf$ \{ r | G $ has a nowhere-zero $ r $-flow $ \} $, and it is denoted by $ F_c(G) $.

A graph with maximum vertex degree $ k $ is a class 1 graph if its edge chromatic number is $ k $.

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

Keywords: nowhere-zero flow, edge-colorings, regular graphs

REAL* Free!! Match Masters Coins Cheats Trick 2024 ★★

Author(s):

REAL* Free!! Match Masters Coins Cheats Trick 2024

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World of Warships Cheats Generator 2024 (generator!) ★★

Author(s):

World of Warships Cheats Generator 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

Matching cut and girth ★★

Author(s):

Question   For every $ d $ does there exists a $ g $ such that every graph with average degree smaller than $ d $ and girth at least $ g $ has a matching-cut?

Keywords: matching cut, matching, cut

3-Colourability of Arrangements of Great Circles ★★

Author(s): Felsner; Hurtado; Noy; Streinu

Consider a set $ S $ of great circles on a sphere with no three circles meeting at a point. The arrangement graph of $ S $ has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.

Conjecture   Every arrangement graph of a set of great circles is $ 3 $-colourable.

Keywords: arrangement graph; graph coloring

Weighted colouring of hexagonal graphs. ★★

Author(s): McDiarmid; Reed

Conjecture   There is an absolute constant $ c $ such that for every hexagonal graph $ G $ and vertex weighting $ p:V(G)\rightarrow \mathbb{N} $, $$\chi(G,p) \leq \frac{9}{8}\omega(G,p) + c $$

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Hungry Shark World Cheats Generator 2024 (Legal) ★★

Author(s):

Hungry Shark World Cheats Generator 2024 (Legal)

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Decomposing an even tournament in directed paths. ★★★

Author(s): Alspach; Mason; Pullman

Conjecture   Every tournament $ D $ on an even number of vertices can be decomposed into $ \sum_{v\in V}\max\{0,d^+(v)-d^-(v)\} $ directed paths.

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Algebraic independence of pi and e ★★★

Author(s):

Conjecture   $ \pi $ and $ e $ are algebraically independent

Keywords: algebraic independence

Packing T-joins ★★

Author(s): DeVos

Conjecture   There exists a fixed constant $ c $ (probably $ c=1 $ suffices) so that every graft with minimum $ T $-cut size at least $ k $ contains a $ T $-join packing of size at least $ (2/3)k-c $.

Keywords: packing; T-join

Invariant subspace problem ★★★

Author(s):

Problem   Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?

Keywords: subspace

Arc-disjoint directed cycles in regular directed graphs ★★

Author(s): Alon; McDiarmid; Molloy

Conjecture   If $ G $ is a $ k $-regular directed graph with no parallel arcs, then $ G $ contains a collection of $ {k+1 \choose 2} $ arc-disjoint directed cycles.

Keywords:

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

Sidorenko's Conjecture ★★★

Author(s): Sidorenko

Conjecture   For any bipartite graph $ H $ and graph $ G $, the number of homomorphisms from $ H $ to $ G $ is at least $ \left(\frac{2|E(G)|}{|V(G)|^2}\right)^{|E(H)|}|V(G)|^{|V(H)|} $.

Keywords: density problems; extremal combinatorics; homomorphism

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

Edge Reconstruction Conjecture ★★★

Author(s): Harary

Conjecture  

Every simple graph with at least 4 edges is reconstructible from it's edge deleted subgraphs

Keywords: reconstruction

Sum of prime and semiprime conjecture ★★

Author(s): Geoffrey Marnell

Conjecture   Every even number greater than $ 10 $ can be represented as the sum of an odd prime number and an odd semiprime .

Keywords: prime; semiprime

Lords Mobile Gems Coins Cheats Mod Android Ios No Survey 2024 (NEW) ★★

Author(s):

Lords Mobile Gems Coins Cheats Mod Android Ios No Survey 2024 (NEW)

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Choosability of Graph Powers ★★

Author(s): Noel

Question  (Noel, 2013)   Does there exist a function $ f(k)=o(k^2) $ such that for every graph $ G $, \[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\]

Keywords: choosability; chromatic number; list coloring; square of a graph

3-accessibility of Fibonacci numbers ★★

Author(s): Landman; Robertson

Question   Is the set of Fibonacci numbers 3-accessible?

Keywords: Fibonacci numbers; monochromatic diffsequences