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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:

Jacobian Conjecture ★★★

Author(s): Keller

Conjecture   Let $ k $ be a field of characteristic zero. A collection $ f_1,\ldots,f_n $ of polynomials in variables $ x_1,\ldots,x_n $ defines an automorphism of $ k^n $ if and only if the Jacobian matrix is a nonzero constant.

Keywords: Affine Geometry; Automorphisms; Polynomials

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

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:

Ádám's Conjecture ★★★

Author(s): Ádám

Conjecture   Every digraph with at least one directed cycle has an arc whose reversal reduces the number of directed cycles.

Keywords:

Special Primes

Author(s): George BALAN

Conjecture   Let $ p $ be a prime natural number. Find all primes $ q\equiv1\left(\mathrm{mod}\: p\right) $, such that $ 2^{\frac{\left(q-1\right)}{p}}\equiv1\left(\mathrm{mod}\: q\right) $.

Keywords:

r-regular graphs are not uniquely hamiltonian. ★★★

Author(s): Sheehan

Conjecture   If $ G $ is a finite $ r $-regular graph, where $ r > 2 $, then $ G $ is not uniquely hamiltonian.

Keywords: hamiltonian; regular; uniquely hamiltonian

A homomorphism problem for flows ★★

Author(s): DeVos

Conjecture   Let $ M,M' $ be abelian groups and let $ B \subseteq M $ and $ B' \subseteq M' $ satisfy $ B=-B $ and $ B' = -B' $. If there is a homomorphism from $ Cayley(M,B) $ to $ Cayley(M',B') $, then every graph with a B-flow has a B'-flow.

Keywords: homomorphism; nowhere-zero flow; tension

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

P vs. NP ★★★★

Author(s): Cook; Levin

Problem   Is P = NP?

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

3 is a primitive root modulo primes of the form 16 q^4 + 1, where q>3 is prime ★★

Author(s):

Conjecture   $ 3~ $ is a primitive root modulo $ ~p $ for all primes $ ~p=16\cdot q^4+1 $, where $ ~q>3 $ is prime.

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

Universal point sets for planar graphs ★★★

Author(s): Mohar

We say that a set $ P \subseteq {\mathbb R}^2 $ is $ n $-universal if every $ n $ vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in $ P $, and all edges are (non-intersecting) straight line segments.

Question   Does there exist an $ n $-universal set of size $ O(n) $?

Keywords: geometric graph; planar graph; universal set

Subgroup formed by elements of order dividing n ★★

Author(s): Frobenius

Conjecture  

Suppose $ G $ is a finite group, and $ n $ is a positive integer dividing $ |G| $. Suppose that $ G $ has exactly $ n $ solutions to $ x^{n} = 1 $. Does it follow that these solutions form a subgroup of $ G $?

Keywords: order, dividing

Acyclic edge-colouring ★★

Author(s): Fiamcik

Conjecture   Every simple graph with maximum degree $ \Delta $ has a proper $ (\Delta+2) $-edge-colouring so that every cycle contains edges of at least three distinct colours.

Keywords: edge-coloring

Wide partition conjecture ★★

Author(s): Chow; Taylor

Conjecture   An integer partition is wide if and only if it is Latin.

Keywords:

Weighted colouring of hexagonal graphs. ★★

Author(s): McDiarmid; Reed

Conjecture   There is an absolute constant $ c $ such that for every hexagonal graph $ G $ and vertex weighting $ p:V(G)\rightarrow \mathbb{N} $, $$\chi(G,p) \leq \frac{9}{8}\omega(G,p) + c $$

Keywords:

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

Inscribed Square Problem ★★

Author(s): Toeplitz

Conjecture   Does every Jordan curve have 4 points on it which form the vertices of a square?

Keywords: simple closed curve; square

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

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

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

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

Aharoni-Berger conjecture ★★★

Author(s): Aharoni; Berger

Conjecture   If $ M_1,\ldots,M_k $ are matroids on $ E $ and $ \sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1) $ for every partition $ \{X_1,\ldots,X_k\} $ of $ E $, then there exists $ X \subseteq E $ with $ |X| = \ell $ which is independent in every $ M_i $.

Keywords: independent set; matroid; partition

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

Consecutive non-orientable embedding obstructions ★★★

Author(s):

Conjecture   Is there a graph $ G $ that is a minor-minimal obstruction for two non-orientable surfaces?

Keywords: minor; surface

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

Dividing up the unrestricted partitions ★★

Author(s): David S.; Newman

Begin with the generating function for unrestricted partitions:

(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...

Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.

Keywords: congruence properties; partition

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

Durer's Conjecture ★★★

Author(s): Durer; Shephard

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

Keywords: folding; polytope

Closing Lemma for Diffeomorphism (Dynamical Systems) ★★★★

Author(s): Charles Pugh

Conjecture   Let $ f\in Diff^{r}(M) $ and $ p\in\omega_{f}  $. Then for any neighborhood $ V_{f}\subset Diff^{r}(M)  $ there is $ g\in V_{f} $ such that $ p $ is periodic point of $ g $

There is an analogous conjecture for flows ( $ C^{r} $ vector fields . In the case of diffeos this was proved by Charles Pugh for $ r = 1 $. In the case of Flows this has been solved by Sushei Hayahshy for $ r = 1 $ . But in the two cases the problem is wide open for $ r > 1 $

Keywords: Dynamics , Pertubation

Diagonal Ramsey numbers ★★★★

Author(s): Erdos

Let $ R(k,k) $ denote the $ k^{th} $ diagonal Ramsey number.

Conjecture   $ \lim_{k \rightarrow \infty} R(k,k) ^{\frac{1}{k}} $ exists.
Problem   Determine the limit in the above conjecture (assuming it exists).

Keywords: Ramsey number

Outward reloid of composition vs composition of outward reloids ★★

Author(s): Porton

Conjecture   For every composable funcoids $ f $ and $ g $ $$(\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq(\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{RLD})_{\mathrm{out}}f.$$

Keywords: outward reloid

Coloring random subgraphs ★★

Author(s): Bukh

If $ G $ is a graph and $ p \in [0,1] $, we let $ G_p $ denote a subgraph of $ G $ where each edge of $ G $ appears in $ G_p $ with independently with probability $ p $.

Problem   Does there exist a constant $ c $ so that $ {\mathbb E}(\chi(G_{1/2})) > c \frac{\chi(G)}{\log \chi(G)} $?

Keywords: coloring; random graph

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

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:

Graphs of exact colorings ★★

Author(s):

Conjecture For $  c \geq m \geq 1  $, let $  P(c,m)  $ be the statement that given any exact $  c  $-coloring of the edges of a complete countably infinite graph (that is, a coloring with $  c  $ colors all of which must be used at least once), there exists an exactly $  m  $-colored countably infinite complete subgraph. Then $  P(c,m)  $ is true if and only if $  m=1  $, $  m=2  $, or $  c=m  $.

Keywords:

Graceful Tree Conjecture ★★★

Author(s):

Conjecture   All trees are graceful

Keywords: combinatorics; graceful labeling

Rota's unimodal conjecture ★★★

Author(s): Rota

Let $ M $ be a matroid of rank $ r $, and for $ 0 \le i \le r $ let $ w_i $ be the number of closed sets of rank $ i $.

Conjecture   $ w_0,w_1,\ldots,w_r $ is unimodal.
Conjecture   $ w_0,w_1,\ldots,w_r $ is log-concave.

Keywords: flat; log-concave; matroid

Hamiltonian paths and cycles in vertex transitive graphs ★★★

Author(s): Lovasz

Problem   Does every connected vertex-transitive graph have a Hamiltonian path?

Keywords: cycle; hamiltonian; path; vertex-transitive

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

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

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

Beneš Conjecture ★★★

Author(s): Beneš

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

Problem  ($ \star $)   Find a sufficient condition for a sequence of partitions $ {\bf h}_1, \dots, {\bf h}_\ell $ of $ E $ to be universal, i.e. to yield the following decomposition of the symmetric group $ \frak S(E) $ on $ E $: $$ (1)\quad \frak S(E) = S({\bf h}_1) S({\bf h}_2) \dots S({\bf h}_\ell).  $$ In particular, what about the sequence $ \bf h,\delta(\bf h),\dots,\delta^{\ell-1}(\bf h) $, where $ \delta $ is a permutation 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:

Roller Coaster permutations ★★★

Author(s): Ahmed; Snevily

Let $ S_n $ denote the set of all permutations of $ [n]=\set{1,2,\ldots,n} $. Let $ i(\pi) $ and $ d(\pi) $ denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in $ \pi $. Let $ X(\pi) $ denote the set of subsequences of $ \pi $ with length at least three. Let $ t(\pi) $ denote $ \sum_{\tau\in X(\pi)}(i(\tau)+d(\tau)) $.

A permutation $ \pi\in S_n $ is called a Roller Coaster permutation if $ t(\pi)=\max_{\tau\in S_n}t(\tau) $. Let $ RC(n) $ be the set of all Roller Coaster permutations in $ S_n $.

Conjecture   For $ n\geq 3 $,
    \item If $ n=2k $, then $ |RC(n)|=4 $. \item If $ n=2k+1 $, then $ |RC(n)|=2^j $ with $ j\leq k+1 $.
Conjecture  (Odd Sum conjecture)   Given $ \pi\in RC(n) $,
    \item If $ n=2k+1 $, then $ \pi_j+\pi_{n-j+1} $ is odd for $ 1\leq j\leq k $. \item If $ n=2k $, then $ \pi_j + \pi_{n-j+1} = 2k+1 $ for all $ 1\leq j\leq k $.

Keywords:

Inverse Galois Problem ★★★★

Author(s): Hilbert

Conjecture   Every finite group is the Galois group of some finite algebraic extension of $ \mathbb Q $.

Keywords:

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

Earth-Moon Problem ★★

Author(s): Ringel

Problem   What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?

Keywords:

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

Book Thickness of Subdivisions ★★

Author(s): Blankenship; Oporowski

Let $ G $ be a finite undirected simple graph.

A $ k $-page book embedding of $ G $ consists of a linear order $ \preceq $ of $ V(G) $ and a (non-proper) $ k $-colouring of $ E(G) $ such that edges with the same colour do not cross with respect to $ \preceq $. That is, if $ v\prec x\prec w\prec y $ for some edges $ vw,xy\in E(G) $, then $ vw $ and $ xy $ receive distinct colours.

One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.

The book thickness of $ G $, denoted by bt$ (G) $ is the minimum integer $ k $ for which there is a $ k $-page book embedding of $ G $.

Let $ G' $ be the graph obtained by subdividing each edge of $ G $ exactly once.

Conjecture   There is a function $ f $ such that for every graph $ G $, $$   \text{bt}(G) \leq f( \text{bt}(G') )\enspace.   $$

Keywords: book embedding; book thickness