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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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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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Earth-Moon Problem ★★

Author(s): Ringel

Problem   What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?

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Hamilton cycle in small d-diregular graphs ★★

Author(s): Jackson

An directed graph is $ k $-diregular if every vertex has indegree and outdegree at least $ k $.

Conjecture   For $ d >2 $, every $ d $-diregular oriented graph on at most $ 4d+1 $ vertices has a Hamilton cycle.

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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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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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Minimal graphs with a prescribed number of spanning trees ★★

Author(s): Azarija; Skrekovski

Conjecture   Let $ n \geq 3 $ be an integer and let $ \alpha(n) $ denote the least integer $ k $ such that there exists a simple graph on $ k $ vertices having precisely $ n $ spanning trees. Then $  \alpha(n) = o(\log{n}). $

Keywords: number of spanning trees, asymptotics

Monochromatic vertex colorings inherited from Perfect Matchings ★★★

Author(s):

Conjecture   For which values of $ n $ and $ d $ are there bi-colored graphs on $ n $ vertices and $ d $ different colors with the property that all the $ d $ monochromatic colorings have unit weight, and every other coloring cancels out?

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Erdős–Straus conjecture ★★

Author(s): Erdos; Straus

Conjecture  

For all $ n > 2 $, there exist positive integers $ x $, $ y $, $ z $ such that $$1/x + 1/y + 1/z = 4/n$$.

Keywords: Egyptian fraction

Few subsequence sums in Z_n x Z_n ★★

Author(s): Bollobas; Leader

Conjecture   For every $ 0 \le t \le n-1 $, the sequence in $ {\mathbb Z}_n^2 $ consisting of $ n-1 $ copes of $ (1,0) $ and $ t $ copies of $ (0,1) $ has the fewest number of distinct subsequence sums over all zero-free sequences from $ {\mathbb Z}_n^2 $ of length $ n-1+t $.

Keywords: subsequence sum; zero sum

New.updated Kim Kardashian Hollywood Cash Stars Cheats 2024 Free No Verification "Free" ★★

Author(s):

New.updated Kim Kardashian Hollywood Cash Stars Cheats 2024 Free No Verification "Free"

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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

Convex Equipartitions with Extreme Perimeter ★★

Author(s): Nandakumar

To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.

Remark: It appears maximizing the total perimeter is the easier problem.

Keywords: convex equipartition

The circular embedding conjecture ★★★

Author(s): Haggard

Conjecture   Every 2-connected graph may be embedded in a surface so that the boundary of each face is a cycle.

Keywords: cover; cycle

Covering a square with unit squares ★★

Author(s):

Conjecture   For any integer $ n \geq 1 $, it is impossible to cover a square of side greater than $ n $ with $ n^2+1 $ unit squares.

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Square achievement game on an n x n grid ★★

Author(s): Erickson

Problem   Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $ n \times n $ grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides. So the game must result in a winner (the first player) when n=15.

Keywords: game

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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Lindelöf hypothesis ★★

Author(s): Lindelöf

Conjecture   For any $ \epsilon>0 $ $$\zeta\left(\frac12 + it\right) \mbox{ is }\mathcal{O}(t^\epsilon).$$

Since $ \epsilon $ can be replaced by a smaller value, we can also write the conjecture as, for any positive $ \epsilon $, $$\zeta\left(\frac12 + it\right) \mbox{ is }o(t^\varepsilon).$$

Keywords: Riemann Hypothesis; zeta

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

Odd perfect numbers ★★★

Author(s): Ancient/folklore

Conjecture   There is no odd perfect number.

Keywords: perfect number

3-Edge-Coloring Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   Suppose $ G $ with $ |V(G)|>2 $ is a connected cubic graph admitting a $ 3 $-edge coloring. Then there is an edge $ e \in E(G) $ such that the cubic graph homeomorphic to $ G-e $ has a $ 3 $-edge coloring.

Keywords: 3-edge coloring; 4-flow; removable edge

MONOPOLY GO Cheats Generator Unlimited IOS And Android No Survey 2024 (free!!) ★★

Author(s):

MONOPOLY GO Cheats Generator Unlimited IOS And Android No Survey 2024 (free!!)

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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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What is the largest graph of positive curvature?

Author(s): DeVos; Mohar

Problem   What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or antiprism?

Keywords: curvature; planar graph

General position subsets ★★

Author(s): Gowers

Question   What is the least integer $ f(n) $ such that every set of at least $ f(n) $ points in the plane contains $ n $ collinear points or a subset of $ n $ points in general position (no three collinear)?

Keywords: general position subset, no-three-in-line problem

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

Author(s):

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

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Arc-disjoint strongly connected spanning subdigraphs ★★

Author(s): Bang-Jensen; Yeo

Conjecture   There exists an ineteger $ k $ so that every $ k $-arc-connected digraph contains a pair of arc-disjoint strongly connected spanning subdigraphs?

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Family Island Cheats Generator Pro Apk (Android Ios) ★★

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Family Island Cheats Generator Pro Apk (Android Ios)

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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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Dirac's Conjecture ★★

Author(s): Dirac

Conjecture   For every set $ P $ of $ n $ points in the plane, not all collinear, there is a point in $ P $ contained in at least $ \frac{n}{2}-c $ lines determined by $ P $, for some constant $ c $.

Keywords: point set

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

Family Island Cheats Generator 2024 Free No Verification (New.updated) ★★

Author(s):

Family Island Cheats Generator 2024 Free No Verification (New.updated)

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The sum of the two largest eigenvalues (Solved) ★★

Author(s):

The sum of the two largest eigenvalues (Solved)

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8 Ball Pool Free Cash Cheats Link 2024 (that work) ★★

Author(s):

8 Ball Pool Free Cash Cheats Link 2024 (that work)

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The Riemann Hypothesis ★★★★

Author(s): Riemann

The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)

Keywords: Millenium Problems; zeta

The additive basis conjecture ★★★

Author(s): Jaeger; Linial; Payan; Tarsi

Conjecture   For every prime $ p $, there is a constant $ c(p) $ (possibly $ c(p)=p $) so that the union (as multisets) of any $ c(p) $ bases of the vector space $ ({\mathbb Z}_p)^n $ contains an additive basis.

Keywords: additive basis; matrix

Generalised Empty Hexagon Conjecture ★★

Author(s): Wood

Conjecture   For each $ \ell\geq3 $ there is an integer $ f(\ell) $ such that every set of at least $ f(\ell) $ points in the plane contains $ \ell $ collinear points or an empty hexagon.

Keywords: empty hexagon

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

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

Fasted Way! For Free Royal Match Coins Cheats Working 2024 Android Ios ★★

Author(s):

Fasted Way! For Free Royal Match Coins Cheats Working 2024 Android Ios

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Brawlhalla Cheats Generator 2024 No Human Veryfication (codes) ★★

Author(s):

Brawlhalla Cheats Generator 2024 No Human Veryfication (codes)

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Large acyclic induced subdigraph in a planar oriented graph. ★★

Author(s): Harutyunyan

Conjecture   Every planar oriented graph $ D $ has an acyclic induced subdigraph of order at least $ \frac{3}{5} |V(D)| $.

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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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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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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

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

MONOPOLY GO Cheats Generator 2024 (Legal) ★★

Author(s):

MONOPOLY GO Cheats Generator 2024 (Legal)

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

Author(s):

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

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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

Genshin Impact Cheats Generator 2024 Edition Update (WORKS) ★★

Author(s):

Genshin Impact Cheats Generator 2024 Edition Update (WORKS)

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