Random

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

Sum of prime and semiprime conjecture ★★

Author(s): Geoffrey Marnell

Conjecture   Every even number greater than $ 10 $ can be represented as the sum of an odd prime number and an odd semiprime .

Keywords: prime; semiprime

Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

Conjecture   Let $ f $ is a $ T_1 $-separable (the same as $ T_2 $ for symmetric transitive) compact funcoid and $ g $ is a uniform space (reflexive, symmetric, and transitive endoreloid) such that $ ( \mathsf{\tmop{FCD}}) g = f $. Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in this draft article.

Direct proof could be better because with it we would get a little more general statement like this:

Conjecture   Let $ f $ be a $ T_1 $-separable compact reflexive symmetric funcoid and $ g $ be a reloid such that
    \item $ ( \mathsf{\tmop{FCD}}) g = f $; \item $ g \circ g^{- 1} \sqsubseteq g $.

Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity

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.

Keywords:

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

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

Simultaneous partition of hypergraphs ★★

Author(s): Kühn; Osthus

Problem   Let $ H_1 $ and $ H_2 $ be two $ r $-uniform hypergraph on the same vertex set $ V $. Does there always exist a partition of $ V $ into $ r $ classes $ V_1, \dots , V_r $ such that for both $ i=1,2 $, at least $ r!m_i/r^r -o(m_i) $ hyperedges of $ H_i $ meet each of the classes $ V_1, \dots , V_r $?

Keywords:

Refuting random 3SAT-instances on $O(n)$ clauses (weak form) ★★★

Author(s): Feige

Conjecture   For every rational $ \epsilon > 0 $ and every rational $ \Delta $, there is no polynomial-time algorithm for the following problem.

Given is a 3SAT (3CNF) formula $ I $ on $ n $ variables, for some $ n $, and $ m = \floor{\Delta n} $ clauses drawn uniformly at random from the set of formulas on $ n $ variables. Return with probability at least 0.5 (over the instances) that $ I $ is typical without returning typical for any instance with at least $ (1 - \epsilon)m $ simultaneously satisfiable clauses.

Keywords: NP; randomness in TCS; satisfiability

Outer reloid of restricted funcoid ★★

Author(s): Porton

Question   $ ( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f) \cap^{\mathsf{RLD}} ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) $ for every filter objects $ \mathcal{A} $ and $ \mathcal{B} $ and a funcoid $ f\in\mathsf{FCD}(\mathrm{Src}\,f; \mathrm{Dst}\,f) $?

Keywords: direct product of filters; outer reloid

Reed's omega, delta, and chi conjecture ★★★

Author(s): Reed

For a graph $ G $, we define $ \Delta(G) $ to be the maximum degree, $ \omega(G) $ to be the size of the largest clique subgraph, and $ \chi(G) $ to be the chromatic number of $ G $.

Conjecture   $ \chi(G) \le \ceil{\frac{1}{2}(\Delta(G)+1) + \frac{1}{2}\omega(G)} $ for every graph $ G $.

Keywords: coloring

Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems) ★★★★

Author(s):

Conjecture   Let $ Diff^{r}(M)  $ be the space of $ C^{r} $ Diffeomorphisms on the connected , compact and boundaryles manifold M and $ \chi^{r}(M) $ the space of $ C^{r} $ vector fields. There is a dense set $ D\subset Diff^{r}(M) $ ($ D\subset \chi^{r}(M) $ ) such that $ \forall f\in D $ exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space $ M $

This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .

Keywords: Attractors , basins, Finite

Vertex Coloring of graph fractional powers ★★★

Author(s): Iradmusa

Conjecture   Let $ G $ be a graph and $ k $ be a positive integer. The $ k- $power of $ G $, denoted by $ G^k $, is defined on the vertex set $ V(G) $, by connecting any two distinct vertices $ x $ and $ y $ with distance at most $ k $. In other words, $ E(G^k)=\{xy:1\leq d_G(x,y)\leq k\} $. Also $ k- $subdivision of $ G $, denoted by $ G^\frac{1}{k} $, is constructed by replacing each edge $ ij $ of $ G $ with a path of length $ k $. Note that for $ k=1 $, we have $ G^\frac{1}{1}=G^1=G $.
Now we can define the fractional power of a graph as follows:
Let $ G $ be a graph and $ m,n\in \mathbb{N} $. The graph $ G^{\frac{m}{n}} $ is defined by the $ m- $power of the $ n- $subdivision of $ G $. In other words $ G^{\frac{m}{n}}\isdef (G^{\frac{1}{n}})^m $.
Conjecture. Let $ G $ be a connected graph with $ \Delta(G)\geq3 $ and $ m $ be a positive integer greater than 1. Then for any positive integer $ n>m $, we have $ \chi(G^{\frac{m}{n}})=\omega(G^\frac{m}{n}) $.
In [1], it was shown that this conjecture is true in some special cases.

Keywords: chromatic number, fractional power of graph, clique number

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

Sets with distinct subset sums ★★★

Author(s): Erdos

Say that a set $ S \subseteq {\mathbb Z} $ has distinct subset sums if distinct subsets of $ S $ have distinct sums.

Conjecture   There exists a fixed constant $ c $ so that $ |S| \le \log_2(n) + c $ whenever $ S \subseteq \{1,2,\ldots,n\} $ has distinct subset sums.

Keywords: subset sum

Monadic second-order logic with cardinality predicates ★★

Author(s): Courcelle

The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the following atomic formulas:

    \item $ \text{``}\,\mathrm{Card}(X) = \mathrm{Card}(Y)\,\text{''} $ \item $ \text{``}\,\mathrm{Card}(X) \text{ belongs to } A\,\text{''} $

where $ A $ is a fixed recursive set of integers.

Let us fix $ k $ and a closed formula $ F $ in this language.

Conjecture   Is it true that the validity of $ F $ for a graph $ G $ of tree-width at most $ k $ can be tested in polynomial time in the size of $ G $?

Keywords: bounded tree width; cardinality predicates; FMT03-Bedlewo; MSO

Unions of triangle free graphs ★★★

Author(s): Erdos; Hajnal

Problem   Does there exist a graph with no subgraph isomorphic to $ K_4 $ which cannot be expressed as a union of $ \aleph_0 $ triangle free graphs?

Keywords: forbidden subgraph; infinite graph; triangle free

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

List Hadwiger Conjecture ★★

Author(s): Kawarabayashi; Mohar

Conjecture   Every $ K_t $-minor-free graph is $ c t $-list-colourable for some constant $ c\geq1 $.

Keywords: Hadwiger conjecture; list colouring; minors

KPZ Universality Conjecture ★★★

Author(s):

Conjecture   Formulate a central limit theorem for the KPZ universality class.

Keywords: KPZ equation, central limit theorem

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

Counterexamples to the Baillie-PSW primality test ★★

Author(s):

Problem  (1)   Find a counterexample to Baillie-PSW primality test or prove that there is no one.
Problem  (2)   Find a composite $ n\equiv 3 $ or $ 7\pmod{10} $ which divides both $ 2^{n-1} - 1 $ (see Fermat pseudoprime) and the Fibonacci number $ F_{n+1} $ (see Lucas pseudoprime), or prove that there is no such $ n $.

Keywords:

Reconstruction conjecture ★★★★

Author(s): Kelly; Ulam

The deck of a graph $ G $ is the multiset consisting of all unlabelled subgraphs obtained from $ G $ by deleting a vertex in all possible ways (counted according to multiplicity).

Conjecture   If two graphs on $ \ge 3 $ vertices have the same deck, then they are isomorphic.

Keywords: reconstruction

Fundamental group torsion for subsets of Euclidean 3-space ★★

Author(s): Ancient/folklore

Problem   Does there exist a subset of $ \mathbb R^3 $ such that its fundamental group has an element of finite order?

Keywords: subsets of euclidean space; torsion

Point sets with no empty pentagon

Author(s): Wood

Problem   Classify the point sets with no empty pentagon.

Keywords: combinatorial geometry; visibility graph

Switching reconstruction of digraphs ★★

Author(s): Bondy; Mercier

Question   Are there any switching-nonreconstructible digraphs on twelve or more vertices?

Keywords:

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:

Giuga's Conjecture on Primality ★★

Author(s): Giuseppe Giuga

Conjecture   $ p $ is a prime iff $ ~\displaystyle \sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p $

Keywords: primality

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

Concavity of van der Waerden numbers ★★

Author(s): Landman

For $ k $ and $ \ell $ positive integers, the (mixed) van der Waerden number $ w(k,\ell) $ is the least positive integer $ n $ such that every (red-blue)-coloring of $ [1,n] $ admits either a $ k $-term red arithmetic progression or an $ \ell $-term blue arithmetic progression.

Conjecture   For all $ k $ and $ \ell $ with $ k \geq \ell $, $ w(k,\ell) \geq w(k+1,\ell-1) $.

Keywords: arithmetic progression; van der Waerden

57-regular Moore graph? ★★★

Author(s): Hoffman; Singleton

Question   Does there exist a 57-regular graph with diameter 2 and girth 5?

Keywords: cage; Moore graph

Oriented trees in n-chromatic digraphs ★★★

Author(s): Burr

Conjecture   Every digraph with chromatic number at least $ 2k-2 $ contains every oriented tree of order $ k $ as a subdigraph.

Keywords:

Unit vector flows ★★

Author(s): Jain

Conjecture   For every graph $ G $ without a bridge, there is a flow $ \phi : E(G) \rightarrow S^2 = \{ x \in {\mathbb R}^3 : |x| = 1 \} $.

Conjecture   There exists a map $ q:S^2 \rightarrow \{-4,-3,-2,-1,1,2,3,4\} $ so that antipodal points of $ S^2 $ receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.

Keywords: nowhere-zero flow

Simplexity of the n-cube ★★★

Author(s):

Question   What is the minimum cardinality of a decomposition of the $ n $-cube into $ n $-simplices?

Keywords: cube; decomposition; simplex

Partitionning a tournament into k-strongly connected subtournaments. ★★

Author(s): Thomassen

Problem   Let $ k_1, \dots , k_p $ be positve integer Does there exists an integer $ g(k_1, \dots , k_p) $ such that every $ g(k_1, \dots , k_p) $-strong tournament $ T $ admits a partition $ (V_1\dots , V_p) $ of its vertex set such that the subtournament induced by $ V_i $ is a non-trivial $ k_i $-strong for all $ 1\leq i\leq p $.

Keywords:

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?

Keywords:

Turán's problem for hypergraphs ★★

Author(s): Turan

Conjecture   Every simple $ 3 $-uniform hypergraph on $ 3n $ vertices which contains no complete $ 3 $-uniform hypergraph on four vertices has at most $ \frac12 n^2(5n-3) $ hyperedges.
Conjecture   Every simple $ 3 $-uniform hypergraph on $ 2n $ vertices which contains no complete $ 3 $-uniform hypergraph on five vertices has at most $ n^2(n-1) $ hyperedges.

Keywords:

Schanuel's Conjecture ★★★★

Author(s): Schanuel

Conjecture   Given any $ n $ complex numbers $ z_1,...,z_n $ which are linearly independent over the rational numbers $ \mathbb{Q} $, then the extension field $ \mathbb{Q}(z_1,...,z_n,\exp(z_1),...,\exp(z_n)) $ has transcendence degree of at least $ n $ over $ \mathbb{Q} $.

Keywords: algebraic independence

Sequence defined on multisets ★★

Author(s): Erickson

Conjecture   Define a $ 2 \times n $ array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array $ [1; 1] $, the sequence is: $ [1; 1] $ -> $ [1; 2] $ -> $ [1, 2; 1, 1] $ -> $ [1, 2; 3, 1] $ -> $ [1, 2, 3; 2, 1, 1] $ -> $ [1, 2, 3; 3, 2, 1] $ -> $ [1, 2, 3; 2, 2, 2] $ -> $ [1, 2, 3; 1, 4, 1] $ -> $ [1, 2, 3, 4; 3, 1, 1, 1] $ -> $ [1, 2, 3, 4; 4, 1, 2, 1] $ -> $ [1, 2, 3, 4; 3, 2, 1, 2] $ -> $ [1, 2, 3, 4; 2, 3, 2, 1] $, and we now have a fixed point (loop of one array).

The process always results in a loop of 1, 2, or 3 arrays.

Keywords: multiset; sequence

Domination in cubic graphs ★★

Author(s): Reed

Problem   Does every 3-connected cubic graph $ G $ satisfy $ \gamma(G) \le \lceil |G|/3 \rceil $ ?

Keywords: cubic graph; domination

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

Partition of a cubic 3-connected graphs into paths of length 2. ★★

Author(s): Kelmans

Problem   Does every $ 3 $-connected cubic graph on $ 3k $ vertices admit a partition into $ k $ paths of length $ 2 $?

Keywords:

Asymptotic Distribution of Form of Polyhedra ★★

Author(s): Rüdinger

Problem   Consider the set of all topologically inequivalent polyhedra with $ k $ edges. Define a form parameter for a polyhedron as $ \beta:= v/(k+2) $ where $ v $ is the number of vertices. What is the distribution of $ \beta $ for $ k \to \infty $?

Keywords: polyhedral graphs, distribution

Beneš Conjecture ★★★

Author(s): Beneš

Let $ E $ be a non-empty finite set. Given a partition $ \bf h $ of $ E $, the stabilizer of $ \bf h $, denoted $ S(\bf h) $, is the group formed by all permutations of $ E $ preserving each block of $ \mathbf h $.

Problem  ($ \star $)   Find a sufficient condition for a sequence of partitions $ {\bf h}_1, \dots, {\bf h}_\ell $ of $ E $ to be complete, i.e. such that the product of their stabilizers $ S({\bf h}_1) S({\bf h}_2) \dots S({\bf h}_\ell) $ is equal to the whole symmetric group $ \frak S(E) $ on $ E $. In particular, what about completeness of the sequence $ \bf h,\delta(\bf h),\dots,\delta^{\ell-1}(\bf h) $, given a partition $ \bf h $ of $ E $ and a permutation $ \delta $ of $ E $?
Conjecture  (Beneš)   Let $ \bf u $ be a uniform partition of $ E $ and $ \varphi $ be a permutation of $ E $ such that $ \bf u\wedge\varphi(\bf u)=\bf 0 $. Suppose that the set $ \big(\varphi S({\bf u})\big)^{n} $ is transitive, for some integer $ n\ge2 $. Then $$ \frak S(E) = \big(\varphi S({\bf u})\big)^{2n-1}. $$

Keywords:

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

Nearly spanning regular subgraphs ★★★

Author(s): Alon; Mubayi

Conjecture   For every $ \epsilon > 0 $ and every positive integer $ k $, there exists $ r_0 = r_0(\epsilon,k) $ so that every simple $ r $-regular graph $ G $ with $ r \ge r_0 $ has a $ k $-regular subgraph $ H $ with $ |V(H)| \ge (1- \epsilon) |V(G)| $.

Keywords: regular; subgraph

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?

Keywords:

Burnside problem ★★★★

Author(s): Burnside

Conjecture   If a group has $ r $ generators and exponent $ n $, is it necessarily finite?

Keywords:

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 $?

Keywords:

The 3n+1 conjecture ★★★

Author(s): Collatz

Conjecture   Let $ f(n) = 3n+1 $ if $ n $ is odd and $ \frac{n}{2} $ if $ n $ is even. Let $ f(1) = 1 $. Assume we start with some number $ n $ and repeatedly take the $ f $ of the current number. Prove that no matter what the initial number is we eventually reach $ 1 $.

Keywords: integer sequence