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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

Solution to the Lonely Runner Conjecture ★★

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Solution to the Lonely Runner Conjecture

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

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Match Masters Free Coins Cheats 2024 (FREE!)

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

Author(s):

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

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The Crossing Number of the Hypercube ★★

Author(s): Erdos; Guy

The crossing number $ cr(G) $ of $ G $ is the minimum number of crossings in all drawings of $ G $ in the plane.

The $ d $-dimensional (hyper)cube $ Q_d $ is the graph whose vertices are all binary sequences of length $ d $, and two of the sequences are adjacent in $ Q_d $ if they differ in precisely one coordinate.

Conjecture   $ \displaystyle \lim  \frac{cr(Q_d)}{4^d} = \frac{5}{32} $

Keywords: crossing number; hypercube

Matchington Mansion Free Stars Coins Cheats Free Generator 2024 in 5 minutes (successive cheats) ★★

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Matchington Mansion Free Stars Coins Cheats Free Generator 2024 in 5 minutes (successive cheats)

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Odd-cycle transversal in triangle-free graphs ★★

Author(s): Erdos; Faudree; Pach; Spencer

Conjecture   If $ G $ is a simple triangle-free graph, then there is a set of at most $ n^2/25 $ edges whose deletion destroys every odd cycle.

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Simpsons Tapped Out Donuts Cash Cheats 2024 (Ios Android) ★★

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Simpsons Tapped Out Donuts Cash Cheats 2024 (Ios Android)

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8 Ball Pool Free Cash Strategy 2024 (The Legit Method) ★★

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8 Ball Pool Free Cash Strategy 2024 (The Legit Method)

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Monochromatic reachability or rainbow triangles ★★★

Author(s): Sands; Sauer; Woodrow

In an edge-colored digraph, we say that a subgraph is rainbow if all its edges have distinct colors, and monochromatic if all its edges have the same color.

Problem   Let $ G $ be a tournament with edges colored from a set of three colors. Is it true that $ G $ must have either a rainbow directed cycle of length three or a vertex $ v $ so that every other vertex can be reached from $ v $ by a monochromatic (directed) path?

Keywords: digraph; edge-coloring; tournament

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

Free Kim Kardashian Hollywood Cash Stars Cheats Pro Apk 2024 (Android Ios) ★★

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Free Kim Kardashian Hollywood Cash Stars Cheats Pro Apk 2024 (Android Ios)

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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

V-Bucks Generator Unlimited Generator (No Human Verification) ★★

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V-Bucks Generator Unlimited Generator (No Human Verification)

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Free Call Of Duty Mobile Cheats Generator No Human Verification No Survey (Unused) ★★

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Free Call Of Duty Mobile Cheats Generator No Human Verification No Survey (Unused)

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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

Funcoidal products inside an inward reloid ★★

Author(s): Porton

Conjecture   (solved) If $ a \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} b \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f $ then $ a \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} b \subseteq f $ for every funcoid $ f $ and atomic f.o. $ a $ and $ b $ on the source and destination of $ f $ correspondingly.

A stronger conjecture:

Conjecture   If $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f $ then $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} \mathcal{B} \subseteq f $ for every funcoid $ f $ and $ \mathcal{A} \in \mathfrak{F} \left( \ensuremath{\operatorname{Src}}f \right) $, $ \mathcal{B} \in \mathfrak{F} \left( \ensuremath{\operatorname{Dst}}f \right) $.

Keywords: inward reloid

Guide ★★

Author(s):

Guide

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Characterizing (aleph_0,aleph_1)-graphs ★★★

Author(s): Diestel; Leader

Call a graph an $ (\aleph_0,\aleph_1) $-graph if it has a bipartition $ (A,B) $ so that every vertex in $ A $ has degree $ \aleph_0 $ and every vertex in $ B $ has degree $ \aleph_1 $.

Problem   Characterize the $ (\aleph_0,\aleph_1) $-graphs.

Keywords: binary tree; infinite graph; normal spanning tree; set theory

Number of Cliques in Minor-Closed Classes ★★

Author(s): Wood

Question   Is there a constant $ c $ such that every $ n $-vertex $ K_t $-minor-free graph has at most $ c^tn $ cliques?

Keywords: clique; graph; minor

Real roots of the flow polynomial ★★

Author(s): Welsh

Conjecture   All real roots of nonzero flow polynomials are at most 4.

Keywords: flow polynomial; nowhere-zero flow

Lucas Numbers Modulo m ★★

Author(s):

Conjecture   The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the following: 2, 4, 6, 7, 14, 3^k, k >=1.

Keywords: Lucas numbers

Raid Shadow Legends Cheats Generator Android Ios 2024 Cheats Generator (HOT) ★★

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Raid Shadow Legends Cheats Generator Android Ios 2024 Cheats Generator (HOT)

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Rendezvous on a line ★★

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Rendezvous on a line

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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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2-accessibility of primes ★★

Author(s): Landman; Robertson

Question   Is the set of prime numbers 2-accessible?

Keywords: monochromatic diffsequences; primes

Wall-Sun-Sun primes and Fibonacci divisibility ★★

Author(s):

Conjecture   For any prime $ p $, there exists a Fibonacci number divisible by $ p $ exactly once.

Equivalently:

Conjecture   For any prime $ p>5 $, $ p^2 $ does not divide $ F_{p-\left(\frac p5\right)} $ where $ \left(\frac mn\right) $ is the Legendre symbol.

Keywords: Fibonacci; prime

Fishing Clash Cheats Generator 2024 No Verification Android iOS (new method) ★★

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Fishing Clash Cheats Generator 2024 No Verification Android iOS (new method)

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Turán Problem for $10$-Cycles in the Hypercube ★★

Author(s): Erdos

Problem   Bound the extremal number of $ C_{10} $ in the hypercube.

Keywords: cycles; extremal combinatorics; hypercube

Every prism over a 3-connected planar graph is hamiltonian. ★★

Author(s): Kaiser; Král; Rosenfeld; Ryjácek; Voss

Conjecture   If $ G $ is a $ 3 $-connected planar graph, then $ G\square K_2 $ has a Hamilton cycle.

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Fishdom Cheats Generator without verification (Free) ★★

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Fishdom Cheats Generator without verification (Free)

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5-flow conjecture ★★★★

Author(s): Tutte

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

Keywords: cubic; nowhere-zero flow

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

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Family Island Cheats Generator Pro Apk (Android Ios)

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Warframe Cheats Generator (iOS Android 2024) ★★

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Warframe Cheats Generator (iOS Android 2024)

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Strong colorability ★★★

Author(s): Aharoni; Alon; Haxell

Let $ r $ be a positive integer. We say that a graph $ G $ is strongly $ r $-colorable if for every partition of the vertices to sets of size at most $ r $ there is a proper $ r $-coloring of $ G $ in which the vertices in each set of the partition have distinct colors.

Conjecture   If $ \Delta $ is the maximal degree of a graph $ G $, then $ G $ is strongly $ 2 \Delta $-colorable.

Keywords: strong coloring

Graphs with a forbidden induced tree are chi-bounded ★★★

Author(s): Gyarfas

Say that a family $ {\mathcal F} $ of graphs is $ \chi $-bounded if there exists a function $ f: {\mathbb N} \rightarrow {\mathbb N} $ so that every $ G \in {\mathcal F} $ satisfies $ \chi(G) \le f (\omega(G)) $.

Conjecture   For every fixed tree $ T $, the family of graphs with no induced subgraph isomorphic to $ T $ is $ \chi $-bounded.

Keywords: chi-bounded; coloring; excluded subgraph; tree

The Berge-Fulkerson conjecture ★★★★

Author(s): Berge; Fulkerson

Conjecture   If $ G $ is a bridgeless cubic graph, then there exist 6 perfect matchings $ M_1,\ldots,M_6 $ of $ G $ with the property that every edge of $ G $ is contained in exactly two of $ M_1,\ldots,M_6 $.

Keywords: cubic; perfect matching

One-way functions exist ★★★★

Author(s):

Conjecture   One-way functions exist.

Keywords: one way function

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

Author(s):

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

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Fasted Way! For Free Star Stable Star Coins Jorvik Coins Cheats Working 2024 Android Ios ★★

Author(s):

Fasted Way! For Free Star Stable Star Coins Jorvik Coins Cheats Working 2024 Android Ios

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Seymour's Second Neighbourhood Conjecture ★★★

Author(s): Seymour

Conjecture   Any oriented graph has a vertex whose outdegree is at most its second outdegree.

Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour

trace inequality ★★

Author(s):

Let $ A,B $ be positive semidefinite, by Jensen's inequality, it is easy to see $ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $, whenever $ s>r>0 $.

What about the $ tr(A^s+B^s)^{\frac{1}{s}}\leq tr(A^r+B^r)^{\frac{1}{r}} $, is it still valid?

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

Author(s):

Marvel Strike Force Cheats Generator Working (refreshed version)

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Discrete Logarithm Problem ★★★

Author(s):

If $ p $ is prime and $ g,h \in {\mathbb Z}_p^* $, we write $ \log_g(h) = n $ if $ n \in {\mathbb Z} $ satisfies $ g^n =  h $. The problem of finding such an integer $ n $ for a given $ g,h \in {\mathbb Z}^*_p $ (with $ g \neq 1 $) is the Discrete Log Problem.

Conjecture   There does not exist a polynomial time algorithm to solve the Discrete Log Problem.

Keywords: discrete log; NP

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

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

Rendezvous on a line ★★★

Author(s): Alpern

Problem   Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time $ R $ (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?

Keywords: game theory; optimization; rendezvous

Drawing disconnected graphs on surfaces ★★

Author(s): DeVos; Mohar; Samal

Conjecture   Let $ G $ be the disjoint union of the graphs $ G_1 $ and $ G_2 $ and let $ \Sigma $ be a surface. Is it true that every optimal drawing of $ G $ on $ \Sigma $ has the property that $ G_1 $ and $ G_2 $ are disjoint?

Keywords: crossing number; surface

Triangle free strongly regular graphs ★★★

Author(s):

Problem   Is there an eighth triangle free strongly regular graph?

Keywords: strongly regular; triangle free

Rise Of Kingdoms Cheats Generator 2024 Update Hacks (Verified) ★★

Author(s):

Rise Of Kingdoms Cheats Generator 2024 Update Hacks (Verified)

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