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The Ultimate Guide to Gardenscapes Cheats and Hacks: Boost Your Game in 2024 ★★

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Conjecture  

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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

Are vertex minor closed classes chi-bounded? ★★

Author(s): Geelen

Question   Is every proper vertex-minor closed class of graphs chi-bounded?

Keywords: chi-bounded; circle graph; coloring; vertex minor

Geodesic cycles and Tutte's Theorem ★★

Author(s): Georgakopoulos; Sprüssel

Problem   If $ G $ is a $ 3 $-connected finite graph, is there an assignment of lengths $ \ell: E(G) \to \mathb R^+ $ to the edges of $ G $, such that every $ \ell $-geodesic cycle is peripheral?

Keywords: cycle space; geodesic cycles; peripheral cycles

Universal highly arc transitive digraphs ★★★

Author(s): Cameron; Praeger; Wormald

An alternating walk in a digraph is a walk $ v_0,e_1,v_1,\ldots,v_m $ so that the vertex $ v_i $ is either the head of both $ e_i $ and $ e_{i+1} $ or the tail of both $ e_i $ and $ e_{i+1} $ for every $ 1 \le i \le m-1 $. A digraph is universal if for every pair of edges $ e,f $, there is an alternating walk containing both $ e $ and $ f $

Question   Does there exist a locally finite highly arc transitive digraph which is universal?

Keywords: arc transitive; digraph

The stubborn list partition problem ★★

Author(s): Cameron; Eschen; Hoang; Sritharan

Problem   Does there exist a polynomial time algorithm which takes as input a graph $ G $ and for every vertex $ v \in V(G) $ a subset $ \ell(v) $ of $ \{1,2,3,4\} $, and decides if there exists a partition of $ V(G) $ into $ \{A_1,A_2,A_3,A_4\} $ so that $ v \in A_i $ only if $ i \in \ell(v) $ and so that $ A_1,A_2 $ are independent, $ A_4 $ is a clique, and there are no edges between $ A_1 $ and $ A_3 $?

Keywords: list partition; polynomial algorithm

Waring rank of determinant ★★

Author(s): Teitler

Question   What is the Waring rank of the determinant of a $ d \times d $ generic matrix?

For simplicity say we work over the complex numbers. The $ d \times d $ generic matrix is the matrix with entries $ x_{i,j} $ for $ 1 \leq i,j \leq d $. Its determinant is a homogeneous form of degree $ d $, in $ d^2 $ variables. If $ F $ is a homogeneous form of degree $ d $, a power sum expression for $ F $ is an expression of the form $ F = \ell_1^d+\dotsb+\ell_r^d $, the $ \ell_i $ (homogeneous) linear forms. The Waring rank of $ F $ is the least number of terms $ r $ in any power sum expression for $ F $. For example, the expression $ xy = \frac{1}{4}(x+y)^2 - \frac{1}{4}(x-y)^2 $ means that $ xy $ has Waring rank $ 2 $ (it can't be less than $ 2 $, as $ xy \neq \ell_1^2 $).

The $ 2 \times 2 $ generic determinant $ x_{1,1}x_{2,2}-x_{1,2}x_{2,1} $ (or $ ad-bc $) has Waring rank $ 4 $. The Waring rank of the $ 3 \times 3 $ generic determinant is at least $ 14 $ and no more than $ 20 $, see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's "VP versus VNP" problem.

Keywords: Waring rank, determinant

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

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

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

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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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Caccetta-Häggkvist Conjecture ★★★★

Author(s): Caccetta; Häggkvist

Conjecture   Every simple digraph of order $ n $ with minimum outdegree at least $ r $ has a cycle with length at most $ \lceil n/r\rceil $

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A funcoid related to directed topological spaces ★★

Author(s): Porton

Conjecture   Let $ R $ be the complete funcoid corresponding to the usual topology on extended real line $ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $. Let $ \geq $ be the order on this set. Then $ R\sqcap^{\mathsf{FCD}}\mathord{\geq} $ is a complete funcoid.
Proposition   It is easy to prove that $ \langle R\sqcap^{\mathsf{FCD}}\mathord{\geq}\rangle \{x\} $ is the infinitely small right neighborhood filter of point $ x\in[-\infty,+\infty] $.

If proved true, the conjecture then can be generalized to a wider class of posets.

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

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

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Long directed cycles in diregular digraphs ★★★

Author(s): Jackson

Conjecture   Every strong oriented graph in which each vertex has indegree and outdegree at least $ d $ contains a directed cycle of length at least $ 2d+1 $.

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

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Clash of Clans Gems Cheats without verification (Free)

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Dice Dreams Cheats Generator 2024 for Android iOS (REAL Generator) ★★

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Dice Dreams Cheats Generator 2024 for Android iOS (REAL Generator)

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eFootball 2023 Cheats Generator 2024 (WORKING IN 5 SECOND) ★★

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eFootball 2023 Cheats Generator 2024 (WORKING IN 5 SECOND)

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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

Convex uniform 5-polytopes ★★

Author(s):

Problem   Enumerate all convex uniform 5-polytopes.

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Dragon City Generator Cheats 2024 (generator!) ★★

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Dragon City Generator Cheats 2024 (generator!)

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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

Point sets with no empty pentagon

Author(s): Wood

Problem   Classify the point sets with no empty pentagon.

Keywords: combinatorial geometry; visibility graph

The Sims Mobile Cheats Generator Working Android Ios 2024 Cheats Generator (Newly Discovered) ★★

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The Sims Mobile Cheats Generator Working Android Ios 2024 Cheats Generator (Newly Discovered)

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Lords Mobile Cheats Unlimited Gems Coins Generator (No Human Verification) ★★

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Lords Mobile Cheats Unlimited Gems Coins Generator (No Human Verification)

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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

Call Of Duty Mobile Generator Cheats No Human Verification (Without Surveys) ★★

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Call Of Duty Mobile Generator Cheats No Human Verification (Without Surveys)

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Special Primes

Author(s): George BALAN

Conjecture   Let $ p $ be a prime natural number. Find all primes $ q\equiv1\left(\mathrm{mod}\: p\right) $, such that $ 2^{\frac{\left(q-1\right)}{p}}\equiv1\left(\mathrm{mod}\: q\right) $.

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The Erdos-Turan conjecture on additive bases ★★★★

Author(s): Erdos; Turan

Let $ B \subseteq {\mathbb N} $. The representation function $ r_B : {\mathbb N} \rightarrow {\mathbb N} $ for $ B $ is given by the rule $ r_B(k) = \#\{ (i,j) \in B \times B : i + j = k \} $. We call $ B $ an additive basis if $ r_B $ is never $ 0 $.

Conjecture   If $ B $ is an additive basis, then $ r_B $ is unbounded.

Keywords: additive basis; representation function

Yu Gi Oh Duel Links Cheats Generator 2024 No Human Veryfication (codes) ★★

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Yu Gi Oh Duel Links Cheats Generator 2024 No Human Veryfication (codes)

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Chromatic Number of Common Graphs ★★

Author(s): Hatami; Hladký; Kráľ; Norine; Razborov

Question   Do common graphs have bounded chromatic number?

Keywords: common graph

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

Smooth 4-dimensional Schoenflies problem ★★★★

Author(s): Alexander

Problem   Let $ M $ be a $ 3 $-dimensional smooth submanifold of $ S^4 $, $ M $ diffeomorphic to $ S^3 $. By the Jordan-Brouwer separation theorem, $ M $ separates $ S^4 $ into the union of two compact connected $ 4 $-manifolds which share $ M $ as a common boundary. The Schoenflies problem asks, are these $ 4 $-manifolds diffeomorphic to $ D^4 $? ie: is $ M $ unknotted?

Keywords: 4-dimensional; Schoenflies; sphere

Directed path of length twice the minimum outdegree ★★★

Author(s): Thomassé

Conjecture   Every oriented graph with minimum outdegree $ k $ contains a directed path of length $ 2k $.

Keywords:

Slice-ribbon problem ★★★★

Author(s): Fox

Conjecture   Given a knot in $ S^3 $ which is slice, is it a ribbon knot?

Keywords: cobordism; knot; ribbon; slice

3-Decomposition Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   (3-Decomposition Conjecture) Every connected cubic graph $ G $ has a decomposition into a spanning tree, a family of cycles and a matching.

Keywords: cubic graph

End-Devouring Rays

Author(s): Georgakopoulos

Problem   Let $ G $ be a graph, $ \omega $ a countable end of $ G $, and $ K $ an infinite set of pairwise disjoint $ \omega $-rays in $ G $. Prove that there is a set $ K' $ of pairwise disjoint $ \omega $-rays that devours $ \omega $ such that the set of starting vertices of rays in $ K' $ equals the set of starting vertices of rays in $ K $.

Keywords: end; ray

"New Cheats" Star Stable Star Coins Jorvik Coins Cheats Free 2024 ★★

Author(s):

"New Cheats" Star Stable Star Coins Jorvik Coins Cheats Free 2024

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Universal Steiner triple systems ★★

Author(s): Grannell; Griggs; Knor; Skoviera

Problem   Which Steiner triple systems are universal?

Keywords: cubic graph; Steiner triple system

Nowhere-zero flows ★★

Author(s):

Nowhere-zero flows

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Unit vector flows ★★

Author(s): Jain

Conjecture   For every graph $ G $ without a bridge, there is a flow $ \phi : E(G) \rightarrow S^2 = \{ x \in {\mathbb R}^3 : |x| = 1 \} $.

Conjecture   There exists a map $ q:S^2 \rightarrow \{-4,-3,-2,-1,1,2,3,4\} $ so that antipodal points of $ S^2 $ receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.

Keywords: nowhere-zero flow

Cookie Run Kingdom Cheats Generator Unlimited Cheats Generator (New 2024) ★★

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Cookie Run Kingdom Cheats Generator Unlimited Cheats Generator (New 2024)

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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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Family Island Cheats Generator 2023-2024 (No Human Verification) ★★

Author(s):

Family Island Cheats Generator 2023-2024 (No Human Verification)

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Jurassic Park Builder Cheats Generator 2024 No Human Verification (Real) ★★

Author(s):

Jurassic Park Builder Cheats Generator 2024 No Human Verification (Real)

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Seagull problem ★★★

Author(s): Seymour

Conjecture   Every $ n $ vertex graph with no independent set of size $ 3 $ has a complete graph on $ \ge \frac{n}{2} $ vertices as a minor.

Keywords: coloring; complete graph; minor

Marvel Strike Force Cheats Generator Working (refreshed version) ★★

Author(s):

Marvel Strike Force Cheats Generator Working (refreshed version)

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Dragon Ball Z Dokkan Battle Cheats Generator 2024 Update (FREE) ★★

Author(s):

Dragon Ball Z Dokkan Battle Cheats Generator 2024 Update (FREE)

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Ding's tau_r vs. tau conjecture ★★★

Author(s): Ding

Conjecture   Let $ r \ge 2 $ be an integer and let $ H $ be a minor minimal clutter with $ \frac{1}{r}\tau_r(H) < \tau(H) $. Then either $ H $ has a $ J_k $ minor for some $ k \ge 2 $ or $ H $ has Lehman's property.

Keywords: clutter; covering; MFMC property; packing

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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Cookie Run Kingdom Cheats Generator Android Ios 2024 Cheats Generator (free) ★★

Author(s):

Cookie Run Kingdom Cheats Generator Android Ios 2024 Cheats Generator (free)

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