Recent Activity

Double-critical graph conjecture ★★

Author(s): Erdos; Lovasz

A connected simple graph $ G $ is called double-critical, if removing any pair of adjacent vertexes lowers the chromatic number by two.

Conjecture   $ K_n $ is the only $ n $-chromatic double-critical graph

Keywords: coloring; complete graph

Shuffle-Exchange Conjecture ★★★

Author(s): Beneš; Folklore; Stone

Given integers $ k,n\ge2 $, let $ d(k,n) $ be the smallest integer $ d\ge2 $ such that the symmetric group $ \frak S $ on the set of all words of length $ n $ over a $ k $-letter alphabet can be generated as $ \frak S = (\sigma \frak G)^d:=\sigma\frak G \sigma\frak G \dots \sigma\frak G $ ($ d $ times), where $ \sigma\in \frak S $ is the shuffle permutation defined by $ \sigma(x_1 x_2 \dots x_{n}) = x_2 \dots x_{n} x_1 $, and $ \frak G $ is the exchange group consisting of all permutations in $ \frak S $ preserving the first $ n-1 $ letters in the words.

Problem  (SE)   Evaluate $ d(k,n) $.
Conjecture  (SE)   $ d(k,n)=2n-1 $, for all $ k,n\ge2 $.


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

Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere? ★★★

Author(s): Novikov

Problem   Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalent to the 4-sphere?

Keywords: 4-sphere; algorithm

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

Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere? ★★★

Author(s): Kirby

Problem   Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.

Keywords: 3-manifold; 4-sphere; embedding

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

Which homology 3-spheres bound homology 4-balls? ★★★★

Author(s): Ancient/folklore

Problem   Is there a complete and computable set of invariants that can determine which (rational) homology $ 3 $-spheres bound (rational) homology $ 4 $-balls?

Keywords: cobordism; homology ball; homology sphere

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

Slice-ribbon problem ★★★★

Author(s): Fox

Conjecture   Given a knot in $ S^3 $ which is slice, is it a ribbon knot?

Keywords: cobordism; knot; ribbon; slice

Smooth 4-dimensional Schoenflies problem ★★★★

Author(s): Alexander

Problem   Let $ M $ be a $ 3 $-dimensional smooth submanifold of $ S^4 $, $ M $ diffeomorphic to $ S^3 $. By the Jordan-Brouwer separation theorem, $ M $ separates $ S^4 $ into the union of two compact connected $ 4 $-manifolds which share $ M $ as a common boundary. The Schoenflies problem asks, are these $ 4 $-manifolds diffeomorphic to $ D^4 $? ie: is $ M $ unknotted?

Keywords: 4-dimensional; Schoenflies; sphere

Are different notions of the crossing number the same? ★★★

Author(s): Pach; Tóth

Problem   Does the following equality hold for every graph $ G $? \[ \text{pair-cr}(G) = \text{cr}(G) \]

The crossing number $ \text{cr}(G) $ of a graph $ G $ is the minimum number of edge crossings in any drawing of $ G $ in the plane. In the pairwise crossing number $ \text{pair-cr}(G) $, we minimize the number of pairs of edges that cross.

Keywords: crossing number; pair-crossing number

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 $.


Edge-Colouring Geometric Complete Graphs ★★

Author(s): Hurtado

Question   What is the minimum number of colours such that every complete geometric graph on $ n $ vertices has an edge colouring such that:
    \item[Variant A] crossing edges get distinct colours, \item[Variant B] disjoint edges get distinct colours, \item[Variant C] non-disjoint edges get distinct colours, \item[Variant D] non-crossing edges get distinct colours.

Keywords: geometric complete graph, colouring

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

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

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

Crossing numbers and coloring ★★★

Author(s): Albertson

We let $ cr(G) $ denote the crossing number of a graph $ G $.

Conjecture   Every graph $ G $ with $ \chi(G) \ge t $ satisfies $ cr(G) \ge cr(K_t) $.

Keywords: coloring; complete graph; crossing number

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

A generalization of Vizing's Theorem? ★★

Author(s): Rosenfeld

Conjecture   Let $ H $ be a simple $ d $-uniform hypergraph, and assume that every set of $ d-1 $ points is contained in at most $ r $ edges. Then there exists an $ r+d-1 $-edge-coloring so that any two edges which share $ d-1 $ vertices have distinct colors.

Keywords: edge-coloring; hypergraph; Vizing