Random

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

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

Author(s):

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

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Vertex Cover Integrality Gap ★★

Author(s): Atserias

Conjecture   For every $ \varepsilon > 0 $ there is $ \delta > 0 $ such that, for every large $ n $, there are $ n $-vertex graphs $ G $ and $ H $ such that $ G \equiv_{\delta n}^{\mathrm{C}} H $ and $ \mathrm{vc}(G) \ge (2 - \varepsilon) \cdot \mathrm{vc}(H) $.

Keywords: counting quantifiers; FMT12-LesHouches

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

Point sets with no empty pentagon

Author(s): Wood

Problem   Classify the point sets with no empty pentagon.

Keywords: combinatorial geometry; visibility graph

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

Cooking Fever Cheats Generator Free 2024 in 5 minutes (New Cheats Generator Cooking Fever) ★★

Author(s):

Cooking Fever Cheats Generator Free 2024 in 5 minutes (New Cheats Generator Cooking Fever)

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Edge-Colouring Geometric Complete Graphs ★★

Author(s): Hurtado

Question   What is the minimum number of colours such that every complete geometric graph on $ n $ vertices has an edge colouring such that:
    \item[Variant A] crossing edges get distinct colours, \item[Variant B] disjoint edges get distinct colours, \item[Variant C] non-disjoint edges get distinct colours, \item[Variant D] non-crossing edges get distinct colours.

Keywords: geometric complete graph, colouring

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

Hedetniemi's Conjecture ★★★

Author(s): Hedetniemi

Conjecture   If $ G,H $ are simple finite graphs, then $ \chi(G \times H) = \min \{ \chi(G), \chi(H) \} $.

Here $ G \times H $ is the tensor product (also called the direct or categorical product) of $ G $ and $ H $.

Keywords: categorical product; coloring; homomorphism; tensor product

Perfect cuboid ★★

Author(s):

Conjecture   Does a perfect cuboid exist?

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Decomposing a connected graph into paths. ★★★

Author(s): Gallai

Conjecture   Every simple connected graph on $ n $ vertices can be decomposed into at most $ \frac{1}{2}(n+1) $ paths.

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Frobenius number of four or more integers ★★

Author(s):

Problem   Find an explicit formula for Frobenius number $ g(a_1, a_2, \dots, a_n) $ of co-prime positive integers $ a_1, a_2, \dots, a_n $ for $ n\geq 4 $.

Keywords:

5-flow conjecture ★★★★

Author(s): Tutte

Conjecture   Every bridgeless graph has a nowhere-zero 5-flow.

Keywords: cubic; nowhere-zero flow

Royal Match Free Coins Cheats 2024 Real Working New Method ★★

Author(s):

Royal Match Free Coins Cheats 2024 Real Working New Method

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Fortnite Working Generator V-Bucks Generator (NEW AND FREE) ★★

Author(s):

Fortnite Working Generator V-Bucks Generator (NEW AND FREE)

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Decomposition of completions of reloids ★★

Author(s): Porton

Conjecture   For composable reloids $ f $ and $ g $ it holds
    \item $ \operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f $ if $ f $ is a co-complete reloid; \item $ \operatorname{CoCompl} ( f \circ g) = f \circ \operatorname{CoCompl} g $ if $ f $ is a complete reloid; \item $ \operatorname{CoCompl} ( ( \operatorname{Compl} g) \circ f) = \operatorname{Compl} ( g \circ   ( \operatorname{CoCompl} f)) = ( \operatorname{Compl} g) \circ ( \operatorname{CoCompl} f) $; \item $ \operatorname{Compl} ( g \circ ( \operatorname{Compl} f)) = \operatorname{Compl} ( g \circ   f) $; \item $ \operatorname{CoCompl} ( ( \operatorname{CoCompl} g) \circ f) = \operatorname{CoCompl} ( g   \circ f) $.

Keywords: co-completion; completion; reloid

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

Bases of many weights ★★★

Author(s): Schrijver; Seymour

Let $ G $ be an (additive) abelian group, and for every $ S \subseteq G $ let $ {\mathit stab}(S) = \{ g \in G : g + S = S \} $.

Conjecture   Let $ M $ be a matroid on $ E $, let $ w : E \rightarrow G $ be a map, put $ S = \{ \sum_{b \in B} w(b) : B \mbox{ is a base} \} $ and $ H = {\mathit stab}(S) $. Then $$|S| \ge |H| \left( 1 - rk(M) + \sum_{Q \in G/H} rk(w^{-1}(Q)) \right).$$

Keywords: matroid; sumset; zero sum

Large induced forest in a planar graph. ★★

Author(s): Abertson; Berman

Conjecture   Every planar graph on $ n $ verices has an induced forest with at least $ n/2 $ vertices.

Keywords:

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

Author(s):

Call Of Duty Mobile Cheats Generator 2024 (LEGIT)

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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

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

Author(s):

Fasted Way! For Free Golf Battle Cheats Generator Working 2024 Android Ios

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Dragon City Cheats Generator 2023-2024 Edition (Verified) ★★

Author(s):

Dragon City Cheats Generator 2023-2024 Edition (Verified)

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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

The Two Color Conjecture ★★

Author(s): Neumann-Lara

Conjecture   If $ G $ is an orientation of a simple planar graph, then there is a partition of $ V(G) $ into $ \{X_1,X_2\} $ so that the graph induced by $ X_i $ is acyclic for $ i=1,2 $.

Keywords: acyclic; digraph; planar

Linear Hypergraphs with Dimension 3 ★★

Author(s): de Fraysseix; Ossona de Mendez; Rosenstiehl

Conjecture   Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.

Keywords: Hypergraphs

Forcing a $K_6$-minor ★★

Author(s): Barát ; Joret; Wood

Conjecture   Every graph with minimum degree at least 7 contains a $ K_6 $-minor.
Conjecture   Every 7-connected graph contains a $ K_6 $-minor.

Keywords: connectivity; graph minors

Melnikov's valency-variety problem

Author(s): Melnikov

Problem   The valency-variety $ w(G) $ of a graph $ G $ is the number of different degrees in $ G $. Is the chromatic number of any graph $ G $ with at least two vertices greater than $$\ceil{ \frac{\floor{w(G)/2}}{|V(G)| - w(G)} } ~ ?$$

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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

Brawlhalla Cheats Generator 2024 (safe and working) ★★

Author(s):

Brawlhalla Cheats Generator 2024 (safe and working)

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P vs. BPP ★★★

Author(s): Folklore

Conjecture   Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a deterministic Turing machine in polynomial time? That is, does P = BPP?

Keywords: BPP; circuit complexity; pseudorandom generators

P vs. NP ★★★★

Author(s): Cook; Levin

Problem   Is P = NP?

Keywords: Complexity Class; Computational Complexity; Millenium Problems; NP; P; polynomial algorithm

Match Masters Free Coins Cheats 2024 (FREE!) ★★

Author(s):

Match Masters Free Coins Cheats 2024 (FREE!)

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Shuffle-Exchange Conjecture (graph-theoretic form) ★★★

Author(s): Beneš; Folklore; Stone

Given integers $ k,n \ge 2 $, the 2-stage Shuffle-Exchange graph/network, denoted $ \text{SE}(k,n) $, is the simple $ k $-regular bipartite graph with the ordered pair $ (U,V) $ of linearly labeled parts $ U:=\{u_0,\dots,u_{t-1}\} $ and $ V:=\{v_0,\dots,v_{t-1}\} $, where $ t:=k^{n-1} $, such that vertices $ u_i $ and $ v_j $ are adjacent if and only if $ (j - ki) \text{ mod } t < k $ (see Fig.1).

Given integers $ k,n,r \ge 2 $, the $ r $-stage Shuffle-Exchange graph/network, denoted $ (\text{SE}(k,n))^{r-1} $, is the proper (i.e., respecting all the orders) concatenation of $ r-1 $ identical copies of $ \text{SE}(k,n) $ (see Fig.1).

Let $ r(k,n) $ be the smallest integer $ r\ge 2 $ such that the graph $ (\text{SE}(k,n))^{r-1} $ is rearrangeable.

Problem   Find $ r(k,n) $.
Conjecture   $ r(k,n)=2n-1 $.

Keywords:

Algebraic independence of pi and e ★★★

Author(s):

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

Keywords: algebraic independence

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

Author(s):

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

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Working Generator Boom Beach Diamonds Cheats Android Ios 2024 (HOT) ★★

Author(s):

Working Generator Boom Beach Diamonds Cheats Android Ios 2024 (HOT)

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Warframe Cheats Generator (iOS Android 2024) ★★

Author(s):

Warframe Cheats Generator (iOS Android 2024)

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Circular choosability of planar graphs

Author(s): Mohar

Let $ G = (V, E) $ be a graph. If $ p $ and $ q $ are two integers, a $ (p,q) $-colouring of $ G $ is a function $ c $ from $ V $ to $ \{0,\dots,p-1\} $ such that $ q \le |c(u)-c(v)| \le p-q $ for each edge $ uv\in E $. Given a list assignment $ L $ of $ G $, i.e.~a mapping that assigns to every vertex $ v $ a set of non-negative integers, an $ L $-colouring of $ G $ is a mapping $ c : V \to N $ such that $ c(v)\in L(v) $ for every $ v\in V $. A list assignment $ L $ is a $ t $-$ (p,q) $-list-assignment if $ L(v) \subseteq \{0,\dots,p-1\} $ and $ |L(v)| \ge tq $ for each vertex $ v \in V $ . Given such a list assignment $ L $, the graph G is $ (p,q) $-$ L $-colourable if there exists a $ (p,q) $-$ L $-colouring $ c $, i.e. $ c $ is both a $ (p,q) $-colouring and an $ L $-colouring. For any real number $ t \ge 1 $, the graph $ G $ is $ t $-$ (p,q) $-choosable if it is $ (p,q) $-$ L $-colourable for every $ t $-$ (p,q) $-list-assignment $ L $. Last, $ G $ is circularly $ t $-choosable if it is $ t $-$ (p,q) $-choosable for any $ p $, $ q $. The circular choosability (or circular list chromatic number or circular choice number) of G is $$cch(G) := \inf\{t \ge 1 : G \text{ is circularly $t$-choosable}\}.$$

Problem   What is the best upper bound on circular choosability for planar graphs?

Keywords: choosability; circular colouring; planar graphs

Do any three longest paths in a connected graph have a vertex in common? ★★

Author(s): Gallai

Conjecture   Do any three longest paths in a connected graph have a vertex in common?

Keywords:

Beneš Conjecture (graph-theoretic form) ★★★

Author(s): Beneš

Problem  ($ \dag $)   Find a sufficient condition for a straight $ \ell $-stage graph to be rearrangeable. In particular, what about a straight uniform graph?
Conjecture  ($ \diamond $)   Let $ L $ be a simple regular ordered $ 2 $-stage graph. Suppose that the graph $ L^m $ is externally connected, for some $ m\ge1 $. Then the graph $ L^{2m} $ is rearrangeable.

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Cycles in Graphs of Large Chromatic Number ★★

Author(s): Brewster; McGuinness; Moore; Noel

Conjecture   If $ \chi(G)>k $, then $ G $ contains at least $ \frac{(k+1)(k-1)!}{2} $ cycles of length $ 0\bmod k $.

Keywords: chromatic number; cycles

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).

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:

4-flow conjecture ★★★

Author(s): Tutte

Conjecture   Every bridgeless graph with no Petersen minor has a nowhere-zero 4-flow.

Keywords: minor; nowhere-zero flow; Petersen graph

Super Meat Boy Forever Points Cheats No Human Verification (Ios Android) ★★

Author(s):

Super Meat Boy Forever Points Cheats No Human Verification (Ios Android)

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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

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

Author(s):

Gardenscapes Cheats Generator 2024 for Android iOS (updated Generator)

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Order-invariant queries ★★

Author(s): Segoufin

Question  
    \item Does $ {<}\text{-invariant\:FO} = \text{FO} $ hold over graphs of bounded tree-width? \item Is $ {<}\text{-invariant\:FO} $ included in $ \text{MSO} $ over graphs? \item Does $ {<}\text{-invariant\:FO} $ have a 0-1 law? \item Are properties of $ {<}\text{-invariant\:FO} $ Hanf-local? \item Is there a logic (with an effective syntax) that captures $ {<}\text{-invariant\:FO} $?

Keywords: Effective syntax; FMT12-LesHouches; Locality; MSO; Order invariance