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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

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

Strong matchings and covers ★★★

Author(s): Aharoni

Let $ H $ be a hypergraph. A strongly maximal matching is a matching $ F \subseteq E(H) $ so that $ |F' \setminus F| \le |F \setminus F'| $ for every matching $ F' $. A strongly minimal cover is a (vertex) cover $ X \subseteq V(H) $ so that $ |X' \setminus X| \ge |X \setminus X'| $ for every cover $ X' $.

Conjecture   If $ H $ is a (possibly infinite) hypergraph in which all edges have size $ \le k $ for some integer $ k $, then $ H $ has a strongly maximal matching and a strongly minimal cover.

Keywords: cover; infinite graph; matching

Erdős-Posa property for long directed cycles ★★

Author(s): Havet; Maia

Conjecture   Let $ \ell \geq 2 $ be an integer. For every integer $ n\geq 0 $, there exists an integer $ t_n=t_n(\ell) $ such that for every digraph $ D $, either $ D $ has a $ n $ pairwise-disjoint directed cycles of length at least $ \ell $, or there exists a set $ T $ of at most $ t_n $ vertices such that $ D-T $ has no directed cycles of length at least $ \ell $.

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List Total Colouring Conjecture ★★

Author(s): Borodin; Kostochka; Woodall

Conjecture   If $ G $ is the total graph of a multigraph, then $ \chi_\ell(G)=\chi(G) $.

Keywords: list coloring; Total coloring; total graphs

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War Thunder Unlimited Generator Golden Eagles Cheats IOS And Android No Survey 2024 (free!!)

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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

Fat 4-polytopes ★★★

Author(s): Eppstein; Kuperberg; Ziegler

The fatness of a 4-polytope $ P $ is defined to be $ (f_1 + f_2)/(f_0 + f_3) $ where $ f_i $ is the number of faces of $ P $ of dimension $ i $.

Question   Does there exist a fixed constant $ c $ so that every convex 4-polytope has fatness at most $ c $?

Keywords: f-vector; polytope

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

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

Length of surreal product

Author(s): Gonshor

Conjecture   Every surreal number has a unique sign expansion, i.e. function $ f: o\rightarrow \{-, +\} $, where $ o $ is some ordinal. This $ o $ is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $ s $ as $ \ell(s) $.

It is easy to prove that

$$ \ell(s+t) \leq \ell(s)+\ell(t) $$

What about

$$ \ell(s\times t) \leq \ell(s)\times\ell(t) $$

?

Keywords: surreal numbers

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FarmVille 2 Coins Farm Bucks Cheats in a few minutes new 2024 (No Survey)

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Gardenscapes Cheats Generator Free Unlimited Cheats Generator (new codes Generator) ★★

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Gardenscapes Cheats Generator Free Unlimited Cheats Generator (new codes Generator)

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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

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Author(s):

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

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Fixed-point logic with counting ★★

Author(s): Blass

Question   Can either of the following be expressed in fixed-point logic plus counting:
    \item Given a graph, does it have a perfect matching, i.e., a set $ M $ of edges such that every vertex is incident to exactly one edge from $ M $? \item Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant?

Keywords: Capturing PTime; counting quantifiers; Fixed-point logic; FMT03-Bedlewo

Friendly partitions ★★

Author(s): DeVos

A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.

Problem   Is it true that for every $ r $, all but finitely many $ r $-regular graphs have friendly partitions?

Keywords: edge-cut; partition; regular

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

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

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

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

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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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Author(s):

New-mathod! Free Kim Kardashian Hollywood Cash Stars Cheats 2024 (No Human Verification)

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Criterion for boundedness of power series

Author(s): Rüdinger

Question   Give a necessary and sufficient criterion for the sequence $ (a_n) $ so that the power series $ \sum_{n=0}^{\infty} a_n x^n $ is bounded for all $ x \in \mathbb{R} $.

Keywords: boundedness; power series; real analysis

Nowhere-zero flows ★★

Author(s):

Nowhere-zero flows

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Raid Shadow Legends Cheats Generator Unlimited IOS And Android No Survey 2024 (free!!) ★★

Author(s):

Raid Shadow Legends Cheats Generator Unlimited IOS And Android No Survey 2024 (free!!)

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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 $.

Keywords:

PTAS for feedback arc set in tournaments ★★

Author(s): Ailon; Alon

Question   Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments?

Keywords: feedback arc set; PTAS; tournament

Combinatorial covering designs

Author(s): Gordon; Mills; Rödl; Schönheim

A $ (v, k, t) $ covering design, or covering, is a family of $ k $-subsets, called blocks, chosen from a $ v $-set, such that each $ t $-subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by $ C(v, k, t) $.

Problem   Find a closed form, recurrence, or better bounds for $ C(v,k,t) $. Find a procedure for constructing minimal coverings.

Keywords: recreational mathematics

Free Idle Miner Tycoon Cheats Generator No Human Verification No Survey (Unused) ★★

Author(s):

Free Idle Miner Tycoon Cheats Generator No Human Verification No Survey (Unused)

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Partitioning edge-connectivity ★★

Author(s): DeVos

Question   Let $ G $ be an $ (a+b+2) $-edge-connected graph. Does there exist a partition $ \{A,B\} $ of $ E(G) $ so that $ (V,A) $ is $ a $-edge-connected and $ (V,B) $ is $ b $-edge-connected?

Keywords: edge-coloring; edge-connectivity

Graham's conjecture on tree reconstruction ★★

Author(s): Graham

Problem   for every graph $ G $, we let $ L(G) $ denote the line graph of $ G $. Given that $ G $ is a tree, can we determine it from the integer sequence $ |V(G)|, |V(L(G))|, |V(L(L(G)))|, \ldots $?

Keywords: reconstruction; tree

A discrete iteration related to Pierce expansions ★★

Author(s): Shallit

Conjecture   Let $ a > b > 0 $ be integers. Set $ b_1 = b $ and $ b_{i+1} = {a \bmod {b_i}} $ for $ i \geq 0 $. Eventually we have $ b_{n+1} = 0 $; put $ P(a,b) = n $.

Example: $ P(35, 22) = 7 $, since $ b_1 = 22 $, $ b_2 = 13 $, $ b_3 = 9 $, $ b_4 = 8 $, $ b_5 = 3 $, $ b_6 = 2 $, $ b_7 = 1 $, $ b_8 = 0 $.

Prove or disprove: $ P(a,b) = O((\log a)^2) $.

Keywords: Pierce expansions

The Erdös-Hajnal Conjecture ★★★

Author(s): Erdos; Hajnal

Conjecture   For every fixed graph $ H $, there exists a constant $ \delta(H) $, so that every graph $ G $ without an induced subgraph isomorphic to $ H $ contains either a clique or an independent set of size $ |V(G)|^{\delta(H)} $.

Keywords: induced subgraph

List chromatic number and maximum degree of bipartite graphs ★★

Author(s): Alon

Conjecture   There is a constant $ c $ such that the list chromatic number of any bipartite graph $ G $ of maximum degree $ \Delta $ is at most $ c \log \Delta $.

Keywords:

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

Author(s):

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

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Kriesell's Conjecture ★★

Author(s): Kriesell

Conjecture   Let $ G $ be a graph and let $ T\subseteq V(G) $ such that for any pair $ u,v\in T $ there are $ 2k $ edge-disjoint paths from $ u $ to $ v $ in $ G $. Then $ G $ contains $ k $ edge-disjoint trees, each of which contains $ T $.

Keywords: Disjoint paths; edge-connectivity; spanning trees

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

The Bollobás-Eldridge-Catlin Conjecture on graph packing ★★★

Author(s):

Conjecture  (BEC-conjecture)   If $ G_1 $ and $ G_2 $ are $ n $-vertex graphs and $ (\Delta(G_1) + 1) (\Delta(G_2) + 1) < n + 1 $, then $ G_1 $ and $ G_2 $ pack.

Keywords: graph packing

2-accessibility of primes ★★

Author(s): Landman; Robertson

Question   Is the set of prime numbers 2-accessible?

Keywords: monochromatic diffsequences; primes

New War Dragons Free Rubies Cheats 2024 Tested (extra) ★★

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New War Dragons Free Rubies Cheats 2024 Tested (extra)

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Odd incongruent covering systems ★★★

Author(s): Erdos; Selfridge

Conjecture   There is no covering system whose moduli are odd, distinct, and greater than 1.

Keywords: covering system

8 Ball Pool Free Cash Cheats Fully Works No Survey (Cheats) ★★

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8 Ball Pool Free Cash Cheats Fully Works No Survey (Cheats)

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Infinite distributivity of meet over join for a principal funcoid ★★

Author(s): Porton

Conjecture   $ f \sqcap \bigsqcup S = \bigsqcup \langle f \sqcap \rangle^{\ast} S $ for principal funcoid $ f $ and a set $ S $ of funcoids of appropriate sources and destinations.

Keywords: distributivity; principal funcoid

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

Chromatic number of associahedron ★★

Author(s): Fabila-Monroy; Flores-Penaloza; Huemer; Hurtado; Urrutia; Wood

Conjecture   Associahedra have unbounded chromatic number.

Keywords: associahedron, graph colouring, chromatic number

The Hodge Conjecture ★★★★

Author(s): Hodge

Conjecture   Let $ X $ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of $ X $.

Keywords: Hodge Theory; Millenium Problems

New World Of Tanks Blitz Free Gold Credits Cheats 2024 Tested (extra) ★★

Author(s):

New World Of Tanks Blitz Free Gold Credits Cheats 2024 Tested (extra)

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Shuffle-Exchange Conjecture ★★

Author(s):

Shuffle-Exchange Conjecture

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Marvel Strike Force Cheats Generator Android Ios 2024 Cheats Generator (HOT) ★★

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Marvel Strike Force Cheats Generator Android Ios 2024 Cheats Generator (HOT)

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Bleach Brave Souls Cheats Generator No Human Verification (Without Surveys) ★★

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Bleach Brave Souls Cheats Generator No Human Verification (Without Surveys)

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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