Recent Activity
Double-critical graph conjecture ★★
A connected simple graph is called double-critical, if removing any pair of adjacent vertexes lowers the chromatic number by two.
Keywords: coloring; complete graph
Shuffle-Exchange Conjecture ★★★
Author(s): Beneš; Folklore; Stone
Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as ( times), where is the shuffle permutation defined by , and is the exchange group consisting of all permutations in preserving the first letters in the words.
Keywords:
Strong colorability ★★★
Author(s): Aharoni; Alon; Haxell
Let be a positive integer. We say that a graph is strongly -colorable if for every partition of the vertices to sets of size at most there is a proper -coloring of in which the vertices in each set of the partition have distinct colors.
Keywords: strong coloring
Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere? ★★★
Author(s): Novikov
What is the homotopy type of the group of diffeomorphisms of the 4-sphere? ★★★★
Author(s): Smale
Keywords: 4-sphere; diffeomorphisms
Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere? ★★★
Author(s): Kirby
Keywords: 3-manifold; 4-sphere; embedding
Fundamental group torsion for subsets of Euclidean 3-space ★★
Author(s): Ancient/folklore
Keywords: subsets of euclidean space; torsion
Which homology 3-spheres bound homology 4-balls? ★★★★
Author(s): Ancient/folklore
Keywords: cobordism; homology ball; homology sphere
Realisation problem for the space of knots in the 3-sphere ★★
Author(s): Budney
Now let be a hyperbolic link. Assume has the further `Brunnian' property that there exists a component of such that is the unlink. Let be the subgroup of consisting of diffeomorphisms of which preserve together with its orientation, and which preserve the orientation of .
There is a representation given by restricting the diffeomorphism to the . It's known that is always a cyclic group. And is a signed symmetric group -- the wreath product of a symmetric group with .
Problem: What representations can be obtained?
Keywords: knot space; symmetry
Slice-ribbon problem ★★★★
Author(s): Fox
Smooth 4-dimensional Schoenflies problem ★★★★
Author(s): Alexander
Keywords: 4-dimensional; Schoenflies; sphere
Are different notions of the crossing number the same? ★★★
The crossing number of a graph is the minimum number of edge crossings in any drawing of in the plane. In the pairwise crossing number , 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 , the 2-stage Shuffle-Exchange graph/network, denoted , is the simple -regular bipartite graph with the ordered pair of linearly labeled parts and , where , such that vertices and are adjacent if and only if (see Fig.1).
Given integers , the -stage Shuffle-Exchange graph/network, denoted , is the proper (i.e., respecting all the orders) concatenation of identical copies of (see Fig.1).
Let be the smallest integer such that the graph is rearrangeable.
Keywords:
Edge-Colouring Geometric Complete Graphs ★★
Author(s): Hurtado
- \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
A gold-grabbing game ★★
Author(s): Rosenfeld
Setup Fix a tree and for every vertex a non-negative integer which we think of as the amount of gold at .
2-Player game Players alternate turns. On each turn, a player chooses a leaf vertex of the tree, takes the gold at this vertex, and then deletes . The game ends when the tree is empty, and the winner is the player who has accumulated the most gold.
Circular colouring the orthogonality graph ★★
Author(s): DeVos; Ghebleh; Goddyn; Mohar; Naserasr
Let denote the graph with vertex set consisting of all lines through the origin in and two vertices adjacent in if they are perpendicular.
Keywords: circular coloring; geometric graph; orthogonality
Crossing numbers and coloring ★★★
Author(s): Albertson
We let denote the crossing number of a graph .
Keywords: coloring; complete graph; crossing number
Domination in cubic graphs ★★
Author(s): Reed
Keywords: cubic graph; domination
A generalization of Vizing's Theorem? ★★
Author(s): Rosenfeld
Keywords: edge-coloring; hypergraph; Vizing