Random

Complexity of square-root sum ★★

Author(s): Goemans

Question   What is the complexity of the following problem?

Given $ a_1,\dots,a_n; k $, determine whether or not $  \sum_i \sqrt{a_i} \leq k.  $

Keywords: semi-definite programming

Gardenscapes Cheats Generator 2024 for Android iOS (updated Generator) ★★

Author(s):

Gardenscapes Cheats Generator 2024 for Android iOS (updated Generator)

Keywords:

Lovász Path Removal Conjecture ★★

Author(s): Lovasz

Conjecture   There is an integer-valued function $ f(k) $ such that if $ G $ is any $ f(k) $-connected graph and $ x $ and $ y $ are any two vertices of $ G $, then there exists an induced path $ P $ with ends $ x $ and $ y $ such that $ G-V(P) $ is $ k $-connected.

Keywords:

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

Shuffle-Exchange Conjecture ★★

Author(s):

Shuffle-Exchange Conjecture

Keywords:

Complexity of the H-factor problem. ★★

Author(s): Kühn; Osthus

An $ H $-factor in a graph $ G $ is a set of vertex-disjoint copies of $ H $ covering all vertices of $ G $.

Problem  Let $ c $ be a fixed positive real number and $ H $ a fixed graph. Is it NP-hard to determine whether a graph $ G $ on $ n $ vertices and minimum degree $ cn $ contains and $ H $-factor?

Keywords:

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

Author(s): Grunbaum; Nash-Williams

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

Keywords:

Graphs with a forbidden induced tree are chi-bounded ★★★

Author(s): Gyarfas

Say that a family $ {\mathcal F} $ of graphs is $ \chi $-bounded if there exists a function $ f: {\mathbb N} \rightarrow {\mathbb N} $ so that every $ G \in {\mathcal F} $ satisfies $ \chi(G) \le f (\omega(G)) $.

Conjecture   For every fixed tree $ T $, the family of graphs with no induced subgraph isomorphic to $ T $ is $ \chi $-bounded.

Keywords: chi-bounded; coloring; excluded subgraph; tree

Something like Picard for 1-forms ★★

Author(s): Elsner

Conjecture   Let $ D $ be the open unit disk in the complex plane and let $ U_1,\dots,U_n $ be open sets such that $ \bigcup_{j=1}^nU_j=D\setminus\{0\} $. Suppose there are injective holomorphic functions $ f_j : U_j \to \mathbb{C}, $ $ j=1,\ldots,n, $ such that for the differentials we have $ {\rm d}f_j={\rm d}f_k $ on any intersection $ U_j\cap U_k $. Then those differentials glue together to a meromorphic 1-form on $ D $.

Keywords: Essential singularity; Holomorphic functions; Picard's theorem; Residue of 1-form; Riemann surfaces

Realisation problem for the space of knots in the 3-sphere ★★

Author(s): Budney

Problem   Given a link $ L $ in $ S^3 $, let the symmetry group of $ L $ be denoted $ Sym(L) = \pi_0 Diff(S^3,L) $ ie: isotopy classes of diffeomorphisms of $ S^3 $ which preserve $ L $, where the isotopies are also required to preserve $ L $.

Now let $ L $ be a hyperbolic link. Assume $ L $ has the further `Brunnian' property that there exists a component $ L_0 $ of $ L $ such that $ L \setminus L_0 $ is the unlink. Let $ A_L $ be the subgroup of $ Sym(L) $ consisting of diffeomorphisms of $ S^3 $ which preserve $ L_0 $ together with its orientation, and which preserve the orientation of $ S^3 $.

There is a representation $ A_L \to \pi_0 Diff(L \setminus L_0) $ given by restricting the diffeomorphism to the $ L \setminus L_0 $. It's known that $ A_L $ is always a cyclic group. And $ \pi_0 Diff(L \setminus L_0) $ is a signed symmetric group -- the wreath product of a symmetric group with $ \mathbb Z_2 $.

Problem: What representations can be obtained?

Keywords: knot space; symmetry

Toon Blast Cheats Generator Android Ios 2024 Cheats Generator (improved version) ★★

Author(s):

Toon Blast Cheats Generator Android Ios 2024 Cheats Generator (improved version)

Keywords:

Solution to the Lonely Runner Conjecture ★★

Author(s):

Solution to the Lonely Runner Conjecture

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Inequality for square summable complex series ★★

Author(s): Retkes

Conjecture   For all $ \alpha=(\alpha_1,\alpha_2,\ldots)\in l_2(\cal{C}) $ the following inequality holds $$\sum_{n\geq 1}|\alpha_n|^2\geq \frac{6}{\pi^2}\sum_{k\geq0}\bigg| \sum_{l\geq0}\frac{1}{l+1}\alpha_{2^k(2l+1)}\bigg|^2 $$

Keywords: Inequality

(m,n)-cycle covers ★★★

Author(s): Celmins; Preissmann

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

Keywords: cover; cycle

Saturation in the Hypercube ★★

Author(s): Morrison; Noel; Scott

Question   What is the saturation number of cycles of length $ 2\ell $ in the $ d $-dimensional hypercube?

Keywords: cycles; hypercube; minimum saturation; saturation

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

Author(s):

Easy! Unlimited Dragon City Cheats Generator codes (GLITCH)

Keywords:

Yu Gi Oh Duel Links Cheats Generator 2024 (safe and working) ★★

Author(s):

Yu Gi Oh Duel Links Cheats Generator 2024 (safe and working)

Keywords:

Dice Dreams Cheats Generator Free Unlimited Cheats Generator (LATEST) ★★

Author(s):

Dice Dreams Cheats Generator Free Unlimited Cheats Generator (LATEST)

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Critical Ops Cheats 2024 Working (Credits Generator) ★★

Author(s):

Critical Ops Cheats 2024 Working (Credits Generator)

Keywords:

Generalized path-connectedness in proximity spaces ★★

Author(s): Porton

Let $ \delta $ be a proximity.

A set $ A $ is connected regarding $ \delta $ iff $ \forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right) $.

Conjecture   The following statements are equivalent for every endofuncoid $ \mu $ and a set $ U $:
    \item $ U $ is connected regarding $ \mu $. \item For every $ a, b \in U $ there exists a totally ordered set $ P \subseteq   U $ such that $ \min P = a $, $ \max P = b $, and for every partion $ \{ X, Y \} $ of $ P $ into two sets $ X $, $ Y $ such that $ \forall x \in X, y \in Y : x < y $, we have $ X \mathrel{[ \mu]^{\ast}} Y $.

Keywords: connected; connectedness; proximity space

Fishing Clash Cheats Generator IOS Android No Verification 2024 (Tips Strategy) ★★

Author(s):

Fishing Clash Cheats Generator IOS Android No Verification 2024 (Tips Strategy)

Keywords:

Boom Beach Unlimited Generator Diamonds Cheats IOS And Android No Survey 2024 (free!!) ★★

Author(s):

Boom Beach Unlimited Generator Diamonds Cheats IOS And Android No Survey 2024 (free!!)

Keywords:

Fishdom Cheats Generator 2024 Edition Update (WORKS) ★★

Author(s):

Fishdom Cheats Generator 2024 Edition Update (WORKS)

Keywords:

S(S(f)) = S(f) for reloids ★★

Author(s): Porton

Question   $ S(S(f)) = S(f) $ for every endo-reloid $ f $?

Keywords: reloid

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

Author(s):

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

Keywords:

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

Author(s):

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

Keywords:

A nowhere-zero point in a linear mapping ★★★

Author(s): Jaeger

Conjecture   If $ {\mathbb F} $ is a finite field with at least 4 elements and $ A $ is an invertible $ n \times n $ matrix with entries in $ {\mathbb F} $, then there are column vectors $ x,y \in {\mathbb F}^n $ which have no coordinates equal to zero such that $ Ax=y $.

Keywords: invertible; nowhere-zero flow

The intersection of two perfect matchings ★★

Author(s): Macajova; Skoviera

Conjecture   Every bridgeless cubic graph has two perfect matchings $ M_1 $, $ M_2 $ so that $ M_1 \cap M_2 $ does not contain an odd edge-cut.

Keywords: cubic; nowhere-zero flow; perfect matching

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

Author(s):

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

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Rainbow Six Siege Cheats Generator Android Ios No Survey 2024 (Current Version) ★★

Author(s):

Rainbow Six Siege Cheats Generator Android Ios No Survey 2024 (Current Version)

Keywords:

Subdivision of a transitive tournament in digraphs with large outdegree. ★★

Author(s): Mader

Conjecture   For all $ k $ there is an integer $ f(k) $ such that every digraph of minimum outdegree at least $ f(k) $ contains a subdivision of a transitive tournament of order $ k $.

Keywords:

Aharoni-Berger conjecture ★★★

Author(s): Aharoni; Berger

Conjecture   If $ M_1,\ldots,M_k $ are matroids on $ E $ and $ \sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1) $ for every partition $ \{X_1,\ldots,X_k\} $ of $ E $, then there exists $ X \subseteq E $ with $ |X| = \ell $ which is independent in every $ M_i $.

Keywords: independent set; matroid; partition

Mastering Subway Surfers: The Ultimate Guide to Cheats, Hacks, and Generators ★★

Author(s):

Mastering Subway Surfers: The Ultimate Guide to Cheats, Hacks, and Generators

Keywords:

Free Super Meat Boy Forever Cheats No Human Verification No Survey (2024 Method) ★★

Author(s):

Free Super Meat Boy Forever Cheats No Human Verification No Survey (2024 Method)

Keywords:

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

Shannon capacity of the seven-cycle ★★★

Author(s):

Problem   What is the Shannon capacity of $ C_7 $?

Keywords:

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

Author(s):

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

Keywords:

"Working Cheats" Warzone COD points Generator No Human Verification 2024 ★★

Author(s):

"Working Cheats" Warzone COD points Generator No Human Verification 2024

Keywords:

Candy Crush Saga Free Golds Lives Cheats 2024-2024 Edition v9 (Verified) ★★

Author(s):

Candy Crush Saga Free Golds Lives Cheats 2024-2024 Edition v9 (Verified)

Keywords:

Codes Free Royal Match Coins Cheats 2024 No Human Veryfication!!! ★★

Author(s):

Codes Free Royal Match Coins Cheats 2024 No Human Veryfication!!!

Keywords:

Golf Battle Cheats Generator Ios and Android 2024 (Working Generator) ★★

Author(s):

Golf Battle Cheats Generator Ios and Android 2024 (Working Generator)

Keywords:

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

Seagull problem ★★

Author(s):

Seagull problem

Keywords:

Star chromatic index of complete graphs ★★

Author(s): Dvorak; Mohar; Samal

Conjecture   Is it possible to color edges of the complete graph $ K_n $ using $ O(n) $ colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?

Equivalently: is the star chromatic index of $ K_n $ linear in $ n $?

Keywords: complete graph; edge coloring; star coloring

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

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

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 $?

Keywords:

Monochromatic empty triangles ★★★

Author(s):

If $ X \subseteq {\mathbb R}^2 $ is a finite set of points which is 2-colored, an empty triangle is a set $ T \subseteq X $ with $ |T|=3 $ so that the convex hull of $ T $ is disjoint from $ X \setminus T $. We say that $ T $ is monochromatic if all points in $ T $ are the same color.

Conjecture   There exists a fixed constant $ c $ with the following property. If $ X \subseteq {\mathbb R}^2 $ is a set of $ n $ points in general position which is 2-colored, then it has $ \ge cn^2 $ monochromatic empty triangles.

Keywords: empty triangle; general position; ramsey theory

Matchings extend to Hamiltonian cycles in hypercubes ★★

Author(s): Ruskey; Savage

Question   Does every matching of hypercube extend to a Hamiltonian cycle?

Keywords: Hamiltonian cycle; hypercube; matching

Which lattices occur as intervals in subgroup lattices of finite groups? ★★★★

Author(s):

Conjecture  

There exists a finite lattice that is not an interval in the subgroup lattice of a finite group.

Keywords: congruence lattice; finite groups