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Grunbaum's Conjecture ★★★

Author(s): Grunbaum

Conjecture   If $ G $ is a simple loopless triangulation of an orientable surface, then the dual of $ G $ is 3-edge-colorable.

Keywords: coloring; surface

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

Author(s):

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

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Pentagon problem ★★★

Author(s): Nesetril

Question   Let $ G $ be a 3-regular graph that contains no cycle of length shorter than $ g $. Is it true that for large enough~$ g $ there is a homomorphism $ G \to C_5 $?

Keywords: cubic; homomorphism

Golf Battle Free Cheats Generator 999,999k Free 2024 (Free Generator) ★★

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Golf Battle Free Cheats Generator 999,999k Free 2024 (Free Generator)

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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

Alexa's Conjecture on Primality ★★

Author(s): Alexa

Definition   Let $ r_i $ be the unique integer (with respect to a fixed $ p\in\mathbb{N} $) such that

$$(2i+1)^{p-1} \equiv r_i \pmod p ~~\text{ and } ~ 0 \le r_i < p. $$

Conjecture   A natural number $ p \ge 8 $ is a prime iff $$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$

Keywords: primality

War Machines Cheats Free Coins Diamonds 2024 No Verification (Android iOS Mod) ★★

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Conjecture  

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Rendezvous on a line ★★

Author(s):

Rendezvous on a line

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SimCity BuildIt Cheats Generator Free 2024 No Human Verification (New Update) ★★

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SimCity BuildIt Cheats Generator Free 2024 No Human Verification (New Update)

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World of Warships Cheats Generator Free Strategy 2024 (The Legit Method) ★★

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World of Warships Cheats Generator Free Strategy 2024 (The Legit Method)

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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

Exponential Algorithms for Knapsack ★★

Author(s): Lipton

Conjecture  

The famous 0-1 Knapsack problem is: Given $ a_{1},a_{2},\dots,a_{n} $ and $ b $ integers, determine whether or not there are $ 0-1 $ values $ x_{1},x_{2},\dots,x_{n} $ so that $$ \sum_{i=1}^{n} a_{i}x_{i} = b.$$ The best known worst-case algorithm runs in time $ 2^{n/2} $ times a polynomial in $ n $. Is there an algorithm that runs in time $ 2^{n/3} $?

Keywords: Algorithm construction; Exponential-time algorithm; Knapsack

Decomposing k-arc-strong tournament into k spanning strong digraphs ★★

Author(s): Bang-Jensen; Yeo

Conjecture   Every k-arc-strong tournament decomposes into k spanning strong digraphs.

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Chromatic number of $\frac{3}{3}$-power of graph ★★

Author(s):

Let $ G $ be a graph and $ m,n\in \mathbb{N} $. The graph $ G^{\frac{m}{n}} $ is defined to be the $ m $-power of the $ n $-subdivision of $ G $. In other words, $ G^{\frac{m}{n}}=(G^{\frac{1}{n}})^m $.

Conjecture   Let $ G $ be a graph with $ \Delta(G)\geq 2 $. Then $ \chi(G^{\frac{3}{3}})\leq 2\Delta(G)+1 $.

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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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Fishing Clash Cheats Generator Free in 2024 (Premium For Free) ★★

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Fishing Clash Cheats Generator Free in 2024 (Premium For Free)

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eFootball 2023 Cheats Generator IOS Android No Verification 2024 (NEW STRATEGY) ★★

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eFootball 2023 Cheats Generator IOS Android No Verification 2024 (NEW STRATEGY)

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Durer's Conjecture ★★★

Author(s): Durer; Shephard

Conjecture   Every convex polytope has a non-overlapping edge unfolding.

Keywords: folding; polytope

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

Bounding the chromatic number of triangle-free graphs with fixed maximum degree ★★

Author(s): Kostochka; Reed

Conjecture   A triangle-free graph with maximum degree $ \Delta $ has chromatic number at most $ \ceil{\frac{\Delta}{2}}+2 $.

Keywords: chromatic number; girth; maximum degree; triangle free

Call Of Duty Mobile Cheats Generator 2024 (LEGIT) ★★

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Call Of Duty Mobile Cheats Generator 2024 (LEGIT)

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Free Clash of Clans Cheats Gems Generator 2023-2024 ★★

Author(s):

Free Clash of Clans Cheats Gems Generator 2023-2024

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The 4x5 chessboard complex is the complement of a link, which link? ★★

Author(s): David Eppstein

Problem   Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.

Keywords: knot theory, links, chessboard complex

Latest Bingo Blitz Cheats Generator 999K Credits Free 2024 in 5 minutes (Up To) ★★

Author(s):

Latest Bingo Blitz Cheats Generator 999K Credits Free 2024 in 5 minutes (Up To)

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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

Which homology 3-spheres bound homology 4-balls? ★★★★

Author(s): Ancient/folklore

Problem   Is there a complete and computable set of invariants that can determine which (rational) homology $ 3 $-spheres bound (rational) homology $ 4 $-balls?

Keywords: cobordism; homology ball; homology sphere

Half-integral flow polynomial values ★★

Author(s): Mohar

Let $ \Phi(G,x) $ be the flow polynomial of a graph $ G $. So for every positive integer $ k $, the value $ \Phi(G,k) $ equals the number of nowhere-zero $ k $-flows in $ G $.

Conjecture   $ \Phi(G,5.5) > 0 $ for every 2-edge-connected graph $ G $.

Keywords: nowhere-zero flow

Are different notions of the crossing number the same? ★★★

Author(s): Pach; Tóth

Problem   Does the following equality hold for every graph $ G $? \[ \text{pair-cr}(G) = \text{cr}(G) \]

The crossing number $ \text{cr}(G) $ of a graph $ G $ is the minimum number of edge crossings in any drawing of $ G $ in the plane. In the pairwise crossing number $ \text{pair-cr}(G) $, we minimize the number of pairs of edges that cross.

Keywords: crossing number; pair-crossing number

Raid Shadow Legends Generator Cheats Free 2024 in 5 minutes (New Generator Cheats Raid Shadow Legends) ★★

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Raid Shadow Legends Generator Cheats Free 2024 in 5 minutes (New Generator Cheats Raid Shadow Legends)

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Free Warframe Cheats Platinum Generator 2024 (Legal) ★★

Author(s):

Free Warframe Cheats Platinum Generator 2024 (Legal)

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Another conjecture about reloids and funcoids ★★

Author(s): Porton

Definition   $ \square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f $ for reloid $ f $.
Conjecture   $ (\mathsf{RLD})_{\Gamma} f = \square (\mathsf{RLD})_{\mathrm{in}} f $ for every funcoid $ f $.

Note: it is known that $ (\mathsf{RLD})_{\Gamma} f \ne \square (\mathsf{RLD})_{\mathrm{out}} f $ (see below mentioned online article).

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Legal* Free Warzone Cheats COD points Generator No Human Verification 2024 ★★

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Legal* Free Warzone Cheats COD points Generator No Human Verification 2024

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Nonseparating planar continuum ★★

Author(s):

Conjecture   Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?

A set has the fixed point property if every continuous map from it into itself has a fixed point.

Keywords: fixed point

Hamiltonian cycles in line graphs ★★★

Author(s): Thomassen

Conjecture   Every 4-connected line graph is hamiltonian.

Keywords: hamiltonian; line graphs

Critical Ops Cheats 2024 Working (Credits Generator) ★★

Author(s):

Critical Ops Cheats 2024 Working (Credits Generator)

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Petersen coloring conjecture ★★★

Author(s): Jaeger

Conjecture   Let $ G $ be a cubic graph with no bridge. Then there is a coloring of the edges of $ G $ using the edges of the Petersen graph so that any three mutually adjacent edges of $ G $ map to three mutually adjancent edges in the Petersen graph.

Keywords: cubic; edge-coloring; Petersen graph

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

Fasted Way! For Free Brawlhalla Cheats Generator Working 2024 Android Ios ★★

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Fasted Way! For Free Brawlhalla Cheats Generator Working 2024 Android Ios

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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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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

New Update: Sims FreePlay Free Simoleons Life Points and Social Points Cheats 2024 No Human Verification ★★

Author(s):

New Update: Sims FreePlay Free Simoleons Life Points and Social Points Cheats 2024 No Human Verification

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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

3-Decomposition Conjectures ★★

Author(s):

Conjecture  

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Degenerate colorings of planar graphs ★★★

Author(s): Borodin

A graph $ G $ is $ k $-degenerate if every subgraph of $ G $ has a vertex of degree $ \le k $.

Conjecture   Every simple planar graph has a 5-coloring so that for $ 1 \le k \le 4 $, the union of any $ k $ color classes induces a $ (k-1) $-degenerate graph.

Keywords: coloring; degenerate; planar

The Crossing Number of the Complete Graph ★★★

Author(s):

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_n) =   \frac 14 \floor{\frac n2} \floor{\frac{n-1}2} \floor{\frac{n-2}2} \floor{\frac{n-3}2} $

Keywords: complete graph; crossing number

Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by projective plane we mean the set of all lines through the origin in $ \mathbb{R}^3 $.

Definition   Say that a subset $ S $ of the projective plane is octahedral if all lines in $ S $ pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition   Say that a subset $ S $ of the projective plane is weakly octahedral if every set $ S'\subseteq S $ such that $ |S'|=3 $ is octahedral.
Conjecture   Suppose that the projective plane can be partitioned into four sets, say $ S_1,S_2,S_3 $ and $ S_4 $ such that each set $ S_i $ is weakly octahedral. Then each $ S_i $ is octahedral.

Keywords: Partitioning; projective plane

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

Author(s):

Conjecture  

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The three 4-flows conjecture ★★

Author(s): DeVos

Conjecture   For every graph $ G $ with no bridge, there exist three disjoint sets $ A_1,A_2,A_3 \subseteq E(G) $ with $ A_1 \cup A_2 \cup A_3 = E(G) $ so that $ G \setminus A_i $ has a nowhere-zero 4-flow for $ 1 \le i \le 3 $.

Keywords: nowhere-zero flow

Almost all non-Hamiltonian 3-regular graphs are 1-connected ★★

Author(s): Haythorpe

Conjecture   Denote by $ NH(n) $ the number of non-Hamiltonian 3-regular graphs of size $ 2n $, and similarly denote by $ NHB(n) $ the number of non-Hamiltonian 3-regular 1-connected graphs of size $ 2n $.

Is it true that $ \lim\limits_{n \rightarrow \infty} \displaystyle\frac{NHB(n)}{NH(n)} = 1 $?

Keywords: Hamiltonian, Bridge, 3-regular, 1-connected

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