Graceful Tree Conjecture ★★★


Conjecture   All trees are graceful

Keywords: combinatorics; graceful labeling

Termination of the sixth Goodstein Sequence

Author(s): Graham

Question   How many steps does it take the sixth Goodstein sequence to terminate?

Keywords: Goodstein Sequence

Total Colouring Conjecture ★★★

Author(s): Behzad

Conjecture   A total coloring of a graph $ G = (V,E) $ is an assignment of colors to the vertices and the edges of $ G $ such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph $ G $, $ \chi''(G) $, equals the minimum number of colors needed in a total coloring of $ G $. It is an old conjecture of Behzad that for every graph $ G $, the total chromatic number equals the maximum degree of a vertex in $ G $, $ \Delta(G) $ plus one or two. In other words, \[\chi''(G)=\Delta(G)+1\ \ or \ \ \Delta(G)+2.\]

Keywords: Total coloring

Perfect cuboid ★★


Conjecture   Does a perfect cuboid exist?


Cycle double cover conjecture ★★★★

Author(s): Seymour; Szekeres

Conjecture   For every graph with no bridge, there is a list of cycles so that every edge is contained in exactly two.

Keywords: cover; cycle

Hamilton decomposition of prisms over 3-connected cubic planar graphs ★★

Author(s): Alspach; Rosenfeld

Conjecture   Every prism over a $ 3 $-connected cubic planar graph can be decomposed into two Hamilton cycles.


The large sets conjecture ★★★

Author(s): Brown; Graham; Landman

Conjecture   If $ A $ is 2-large, then $ A $ is large.

Keywords: 2-large sets; large sets

Odd perfect numbers ★★★

Author(s): Ancient/folklore

Conjecture   There is no odd perfect number.

Keywords: perfect number

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

Obstacle number of planar graphs

Author(s): Alpert; Koch; Laison

Does there exist a planar graph with obstacle number greater than 1? Is there some $ k $ such that every planar graph has obstacle number at most $ k $?

Keywords: graph drawing; obstacle number; planar graph; visibility graph

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

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

Turán number of a finite family. ★★

Author(s): Erdos; Simonovits

Given a finite family $ {\cal F} $ of graphs and an integer $ n $, the Turán number $ ex(n,{\cal F}) $ of $ {\cal F} $ is the largest integer $ m $ such that there exists a graph on $ n $ vertices with $ m $ edges which contains no member of $ {\cal F} $ as a subgraph.

Conjecture   For every finite family $ {\cal F} $ of graphs there exists an $ F\in {\cal F} $ such that $ ex(n, F ) = O(ex(n, {\cal F})) $ .


Cyclic spanning subdigraph with small cyclomatic number ★★

Author(s): Bondy

Conjecture   Let $ D $ be a digraph all of whose strong components are nontrivial. Then $ D $ contains a cyclic spanning subdigraph with cyclomatic number at most $ \alpha(D) $.


Switching reconstruction of digraphs ★★

Author(s): Bondy; Mercier

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


Laplacian Degrees of a Graph ★★

Author(s): Guo

Conjecture   If $ G $ is a connected graph on $ n $ vertices, then $ c_k(G) \ge d_k(G) $ for $ k = 1, 2, \dots, n-1 $.

Keywords: degree sequence; Laplacian matrix

Arc-disjoint out-branching and in-branching ★★

Author(s): Thomassen

Conjecture   There exists an integer $ k $ such that every $ k $-arc-strong digraph $ D $ with specified vertices $ u $ and $ v $ contains an out-branching rooted at $ u $ and an in-branching rooted at $ v $ which are arc-disjoint.


3-accessibility of Fibonacci numbers ★★

Author(s): Landman; Robertson

Question   Is the set of Fibonacci numbers 3-accessible?

Keywords: Fibonacci numbers; monochromatic diffsequences

Long rainbow arithmetic progressions ★★

Author(s): Fox; Jungic; Mahdian; Nesetril; Radoicic

For $ k\in \mathbb{N} $ let $ T_k $ denote the minimal number $ t\in \mathbb{N} $ such that there is a rainbow $ AP(k) $ in every equinumerous $ t $-coloring of $ \{ 1,2,\ldots ,tn\} $ for every $ n\in \mathbb{N} $

Conjecture   For all $ k\geq 3 $, $ T_k=\Theta (k^2) $.

Keywords: arithmetic progression; rainbow

A nowhere-zero point in a linear mapping ★★★

Author(s): Jaeger

Conjecture   If $ {\mathbb F} $ is a finite field with at least 4 elements and $ A $ is an invertible $ n \times n $ matrix with entries in $ {\mathbb F} $, then there are column vectors $ x,y \in {\mathbb F}^n $ which have no coordinates equal to zero such that $ Ax=y $.

Keywords: invertible; nowhere-zero flow

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

Cube-Simplex conjecture ★★★

Author(s): Kalai

Conjecture   For every positive integer $ k $, there exists an integer $ d $ so that every polytope of dimension $ \ge d $ has a $ k $-dimensional face which is either a simplex or is combinatorially isomorphic to a $ k $-dimensional cube.

Keywords: cube; facet; polytope; simplex

2-colouring a graph without a monochromatic maximum clique ★★

Author(s): Hoang; McDiarmid

Conjecture   If $ G $ is a non-empty graph containing no induced odd cycle of length at least $ 5 $, then there is a $ 2 $-vertex colouring of $ G $ in which no maximum clique is monochromatic.

Keywords: maximum clique; Partitioning

What is the homotopy type of the group of diffeomorphisms of the 4-sphere? ★★★★

Author(s): Smale

Problem   $ Diff(S^4) $ has the homotopy-type of a product space $ Diff(S^4) \simeq \mathbb O_5 \times Diff(D^4) $ where $ Diff(D^4) $ is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of $ Diff(D^4) $.

Keywords: 4-sphere; diffeomorphisms

Directed path of length twice the minimum outdegree ★★★

Author(s): Thomassé

Conjecture   Every oriented graph with minimum outdegree $ k $ contains a directed path of length $ 2k $.


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

Durer's Conjecture ★★★

Author(s): Durer; Shephard

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

Keywords: folding; polytope

Subdivision of a transitive tournament in digraphs with large outdegree. ★★

Author(s): Mader

Conjecture   For all $ k $ there is an integer $ f(k) $ such that every digraph of minimum outdegree at least $ f(k) $ contains a subdivision of a transitive tournament of order $ k $.


Euler-Mascheroni constant ★★★


Question   Is Euler-Mascheroni constant an transcendental number?

Keywords: constant; Euler; irrational; Mascheroni; rational; transcendental

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

Realisation problem for the space of knots in the 3-sphere ★★

Author(s): Budney

Problem   Given a link $ L $ in $ S^3 $, let the symmetry group of $ L $ be denoted $ Sym(L) = \pi_0 Diff(S^3,L) $ ie: isotopy classes of diffeomorphisms of $ S^3 $ which preserve $ L $, where the isotopies are also required to preserve $ L $.

Now let $ L $ be a hyperbolic link. Assume $ L $ has the further `Brunnian' property that there exists a component $ L_0 $ of $ L $ such that $ L \setminus L_0 $ is the unlink. Let $ A_L $ be the subgroup of $ Sym(L) $ consisting of diffeomorphisms of $ S^3 $ which preserve $ L_0 $ together with its orientation, and which preserve the orientation of $ S^3 $.

There is a representation $ A_L \to \pi_0 Diff(L \setminus L_0) $ given by restricting the diffeomorphism to the $ L \setminus L_0 $. It's known that $ A_L $ is always a cyclic group. And $ \pi_0 Diff(L \setminus L_0) $ is a signed symmetric group -- the wreath product of a symmetric group with $ \mathbb Z_2 $.

Problem: What representations can be obtained?

Keywords: knot space; symmetry

The 4x5 chessboard complex is the complement of a link, which link? ★★

Author(s): David Eppstein

Problem   Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.

Keywords: knot theory, links, chessboard complex

Cores of Cayley graphs ★★★★★

Author(s): Samal

Conjecture   Let $ M $ be an abelian group. Is the core of a Cayley graph (on some power of $ M $) a Cayley graph (on some power of $ M $)?

Keywords: Cayley graph; core

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

S(S(f)) = S(f) for reloids ★★

Author(s): Porton

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

Keywords: reloid

Exact colorings of graphs ★★

Author(s): Erickson

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: graph coloring; ramsey theory

Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament ★★

Author(s): Yuster

Conjecture   If $ T $ is a tournament of order $ n $, then it contains $ \left \lceil n(n-1)/6 - n/3\right\rceil $ arc-disjoint transitive subtournaments of order 3.


Every metamonovalued funcoid is monovalued ★★

Author(s): Porton

Conjecture   Every metamonovalued funcoid is monovalued.

The reverse is almost trivial: Every monovalued funcoid is metamonovalued.

Keywords: monovalued

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

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


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

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.


Bounding the on-line choice number in terms of the choice number ★★

Author(s): Zhu

Question   Are there graphs for which $ \text{ch}^{\text{OL}}-\text{ch} $ is arbitrarily large?

Keywords: choosability; list coloring; on-line choosability

Hamiltonicity of Cayley graphs ★★★

Author(s): Rapaport-Strasser

Question   Is every Cayley graph Hamiltonian?


Inequality of the means ★★★


Question   Is is possible to pack $ n^n $ rectangular $ n $-dimensional boxes each of which has side lengths $ a_1,a_2,\ldots,a_n $ inside an $ n $-dimensional cube with side length $ a_1 + a_2 + \ldots a_n $?

Keywords: arithmetic mean; geometric mean; Inequality; packing

Ryser's conjecture ★★★

Author(s): Ryser

Conjecture   Let $ H $ be an $ r $-uniform $ r $-partite hypergraph. If $ \nu $ is the maximum number of pairwise disjoint edges in $ H $, and $ \tau $ is the size of the smallest set of vertices which meets every edge, then $ \tau \le (r-1) \nu $.

Keywords: hypergraph; matching; packing

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

Graphs of exact colorings ★★


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  $.


Mixing Circular Colourings

Author(s): Brewster; Noel

Question   Is $ \mathfrak{M}_c(G) $ always rational?

Keywords: discrete homotopy; graph colourings; mixing

Divisibility of central binomial coefficients ★★

Author(s): Graham

Problem  (1)   Prove that there exist infinitely many positive integers $ n $ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$
Problem  (2)   Prove that there exists only a finite number of positive integers $ n $ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7\cdot 11) = 1.$$


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