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The three 4-flows conjecture ★★

Author(s): DeVos

Conjecture   For every graph $ G $ with no bridge, there exist three disjoint sets $ A_1,A_2,A_3 \subseteq E(G) $ with $ A_1 \cup A_2 \cup A_3 = E(G) $ so that $ G \setminus A_i $ has a nowhere-zero 4-flow for $ 1 \le i \le 3 $.

Keywords: nowhere-zero flow

Chords of longest cycles ★★★

Author(s): Thomassen

Conjecture   If $ G $ is a 3-connected graph, every longest cycle in $ G $ has a chord.

Keywords: chord; connectivity; cycle

Davenport's constant ★★★

Author(s):

For a finite (additive) abelian group $ G $, the Davenport constant of $ G $, denoted $ s(G) $, is the smallest integer $ t $ so that every sequence of elements of $ G $ with length $ \ge t $ has a nontrivial subsequence which sums to zero.

Conjecture   $ s( {\mathbb Z}_n^d) = d(n-1) + 1 $

Keywords: Davenport constant; subsequence sum; zero sum

4-regular 4-chromatic graphs of high girth ★★

Author(s): Grunbaum

Problem   Do there exist 4-regular 4-chromatic graphs of arbitrarily high girth?

Keywords: coloring; girth

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

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

Even vs. odd latin squares ★★★

Author(s): Alon; Tarsi

A latin square is even if the product of the signs of all of the row and column permutations is 1 and is odd otherwise.

Conjecture   For every positive even integer $ n $, the number of even latin squares of order $ n $ and the number of odd latin squares of order $ n $ are different.

Keywords: latin square

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:

The Alon-Tarsi basis conjecture ★★

Author(s): Alon; Linial; Meshulam

Conjecture   If $ B_1,B_2,\ldots B_p $ are invertible $ n \times n $ matrices with entries in $ {\mathbb Z}_p $ for a prime $ p $, then there is a $ n \times (p-1)n $ submatrix $ A $ of $ [B_1 B_2 \ldots B_p] $ so that $ A $ is an AT-base.

Keywords: additive basis; matrix

Upgrading a completary multifuncoid ★★

Author(s): Porton

Let $ \mho $ be a set, $ \mathfrak{F} $ be the set of filters on $ \mho $ ordered reverse to set-theoretic inclusion, $ \mathfrak{P} $ be the set of principal filters on $ \mho $, let $ n $ be an index set. Consider the filtrator $ \left( \mathfrak{F}^n ; \mathfrak{P}^n \right) $.

Conjecture   If $ f $ is a completary multifuncoid of the form $ \mathfrak{P}^n $, then $ E^{\ast} f $ is a completary multifuncoid of the form $ \mathfrak{F}^n $.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords:

Universal Steiner triple systems ★★

Author(s): Grannell; Griggs; Knor; Skoviera

Problem   Which Steiner triple systems are universal?

Keywords: cubic graph; Steiner triple system

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

Olson's Conjecture ★★

Author(s): Olson

Conjecture   If $ a_1,a_2,\ldots,a_{2n-1} $ is a sequence of elements from a multiplicative group of order $ n $, then there exist $ 1 \le j_1 < j_2 \ldots < j_n \le 2n-1 $ so that $ \prod_{i=1}^n a_{j_i} = 1 $.

Keywords: zero sum

Are there an infinite number of lucky primes?

Author(s): Lazarus: Gardiner: Metropolis; Ulam

Conjecture   If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.

Keywords: lucky; prime; seive

A gold-grabbing game ★★

Author(s): Rosenfeld

Setup Fix a tree $ T $ and for every vertex $ v \in V(T) $ a non-negative integer $ g(v) $ which we think of as the amount of gold at $ v $.

2-Player game Players alternate turns. On each turn, a player chooses a leaf vertex $ v $ of the tree, takes the gold at this vertex, and then deletes $ v $. The game ends when the tree is empty, and the winner is the player who has accumulated the most gold.

Problem   Find optimal strategies for the players.

Keywords: game; tree

3-accessibility of Fibonacci numbers ★★

Author(s): Landman; Robertson

Question   Is the set of Fibonacci numbers 3-accessible?

Keywords: Fibonacci numbers; monochromatic diffsequences

Unsolvability of word problem for 2-knot complements ★★★

Author(s): Gordon

Problem   Does there exist a smooth/PL embedding of $ S^2 $ in $ S^4 $ such that the fundamental group of the complement has an unsolvable word problem?

Keywords: 2-knot; Computational Complexity; knot theory

Half-integral flow polynomial values ★★

Author(s): Mohar

Let $ \Phi(G,x) $ be the flow polynomial of a graph $ G $. So for every positive integer $ k $, the value $ \Phi(G,k) $ equals the number of nowhere-zero $ k $-flows in $ G $.

Conjecture   $ \Phi(G,5.5) > 0 $ for every 2-edge-connected graph $ G $.

Keywords: nowhere-zero flow

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

End-Devouring Rays

Author(s): Georgakopoulos

Problem   Let $ G $ be a graph, $ \omega $ a countable end of $ G $, and $ K $ an infinite set of pairwise disjoint $ \omega $-rays in $ G $. Prove that there is a set $ K' $ of pairwise disjoint $ \omega $-rays that devours $ \omega $ such that the set of starting vertices of rays in $ K' $ equals the set of starting vertices of rays in $ K $.

Keywords: end; ray

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

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.

Keywords:

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

P vs. PSPACE ★★★

Author(s): Folklore

Problem   Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?

Keywords: P; PSPACE; separation; unconditional

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

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

S(S(f)) = S(f) for reloids ★★

Author(s): Porton

Question   $ S(S(f)) = S(f) $ for every endo-reloid $ f $?

Keywords: reloid

Circular colouring the orthogonality graph ★★

Author(s): DeVos; Ghebleh; Goddyn; Mohar; Naserasr

Let $ {\mathcal O} $ denote the graph with vertex set consisting of all lines through the origin in $ {\mathbb R}^3 $ and two vertices adjacent in $ {\mathcal O} $ if they are perpendicular.

Problem   Is $ \chi_c({\mathcal O}) = 4 $?

Keywords: circular coloring; geometric graph; orthogonality

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

Circular flow number of regular class 1 graphs ★★

Author(s): Steffen

A nowhere-zero $ r $-flow $ (D(G),\phi) $ on $ G $ is an orientation $ D $ of $ G $ together with a function $ \phi $ from the edge set of $ G $ into the real numbers such that $ 1 \leq |\phi(e)| \leq r-1 $, for all $ e \in E(G) $, and $ \sum_{e \in E^+(v)}\phi(e) = \sum_{e \in E^-(v)}\phi(e), \textrm{ for all } v \in V(G) $. The circular flow number of $ G $ is inf$ \{ r | G $ has a nowhere-zero $ r $-flow $ \} $, and it is denoted by $ F_c(G) $.

A graph with maximum vertex degree $ k $ is a class 1 graph if its edge chromatic number is $ k $.

Conjecture   Let $ t \geq 1 $ be an integer and $ G $ a $ (2t+1) $-regular graph. If $ G $ is a class 1 graph, then $ F_c(G) \leq 2 + \frac{2}{t} $.

Keywords: nowhere-zero flow, edge-colorings, regular graphs

Are vertex minor closed classes chi-bounded? ★★

Author(s): Geelen

Question   Is every proper vertex-minor closed class of graphs chi-bounded?

Keywords: chi-bounded; circle graph; coloring; vertex minor

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

A sextic counterexample to Euler's sum of powers conjecture ★★

Author(s): Euler

Problem   Find six positive integers $ x_1, x_2, \dots, x_6 $ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do not exist.

Keywords:

Negative association in uniform forests ★★

Author(s): Pemantle

Conjecture   Let $ G $ be a finite graph, let $ e,f \in E(G) $, and let $ F $ be the edge set of a forest chosen uniformly at random from all forests of $ G $. Then \[ {\mathbb P}(e \in F \mid f \in F}) \le {\mathbb P}(e \in F) \]

Keywords: forest; negative association

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

Crossing sequences ★★

Author(s): Archdeacon; Bonnington; Siran

Conjecture   Let $ (a_0,a_1,a_2,\ldots,0) $ be a sequence of nonnegative integers which strictly decreases until $ 0 $.

Then there exists a graph that be drawn on a surface with orientable (nonorientable, resp.) genus $ i $ with $ a_i $ crossings, but not with less crossings.

Keywords: crossing number; crossing sequence

Arc-disjoint strongly connected spanning subdigraphs ★★

Author(s): Bang-Jensen; Yeo

Conjecture   There exists an ineteger $ k $ so that every $ k $-arc-connected digraph contains a pair of arc-disjoint strongly connected spanning subdigraphs?

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

Stable set meeting all longest directed paths. ★★

Author(s): Laborde; Payan; Xuong N.H.

Conjecture   Every digraph has a stable set meeting all longest directed paths

Keywords:

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:

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

Twin prime conjecture ★★★★

Author(s):

Conjecture   There exist infinitely many positive integers $ n $ so that both $ n $ and $ n+2 $ are prime.

Keywords: prime; twin prime

The Borodin-Kostochka Conjecture ★★

Author(s): Borodin; Kostochka

Conjecture   Every graph with maximum degree $ \Delta \geq 9 $ has chromatic number at most $ \max\{\Delta-1, \omega\} $.

Keywords:

Covering systems with big moduli ★★

Author(s): Erdos; Selfridge

Problem   Does for every integer $ N $ exist a covering system with all moduli distinct and at least equal to~$ N $?

Keywords: covering system

Strict inequalities for products of filters

Author(s): Porton

Conjecture   $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B}   \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A}   \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} $ for some filter objects $ \mathcal{A} $, $ \mathcal{B} $. Particularly, is this formula true for $ \mathcal{A} = \mathcal{B} = \Delta \cap \uparrow^{\mathbb{R}} \left( 0 ; +   \infty \right) $?

A weaker conjecture:

Conjecture   $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B}   \subset \mathcal{A} \ltimes \mathcal{B} $ for some filter objects $ \mathcal{A} $, $ \mathcal{B} $.

Keywords: filter products

Circular flow numbers of $r$-graphs ★★

Author(s): Steffen

A nowhere-zero $ r $-flow $ (D(G),\phi) $ on $ G $ is an orientation $ D $ of $ G $ together with a function $ \phi $ from the edge set of $ G $ into the real numbers such that $ 1 \leq |\phi(e)| \leq r-1 $, for all $ e \in E(G) $, and $ \sum_{e \in E^+(v)}\phi(e) = \sum_{e \in E^-(v)}\phi(e), \textrm{ for all } v \in V(G) $.

A $ (2t+1) $-regular graph $ G $ is a $ (2t+1) $-graph if $ |\partial_G(X)| \geq 2t+1 $ for every $ X \subseteq V(G) $ with $ |X| $ odd.

Conjecture   Let $ t > 1 $ be an integer. If $ G $ is a $ (2t+1) $-graph, then $ F_c(G) \leq 2 + \frac{2}{t} $.

Keywords: flow conjectures; nowhere-zero flows

Unconditional derandomization of Arthur-Merlin games ★★★

Author(s): Shaltiel; Umans

Problem   Prove unconditionally that $ \mathcal{AM} $ $ \subseteq $ $ \Sigma_2 $.

Keywords: Arthur-Merlin; Hitting Sets; unconditional

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

Weak pentagon problem ★★

Author(s): Samal

Conjecture   If $ G $ is a cubic graph not containing a triangle, then it is possible to color the edges of $ G $ by five colors, so that the complement of every color class is a bipartite graph.

Keywords: Clebsch graph; cut-continuous mapping; edge-coloring; homomorphism; pentagon

Generalised Empty Hexagon Conjecture ★★

Author(s): Wood

Conjecture   For each $ \ell\geq3 $ there is an integer $ f(\ell) $ such that every set of at least $ f(\ell) $ points in the plane contains $ \ell $ collinear points or an empty hexagon.

Keywords: empty hexagon