Problem Given a link in , let the symmetry group of be denoted ie: isotopy classes of diffeomorphisms of which preserve , where the isotopies are also required to preserve .
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 .
Conjecture For every positive integer , there exists an integer so that every polytope of dimension has a -dimensional face which is either a simplex or is combinatorially isomorphic to a -dimensional cube.
Let be a simple graph, and for every list assignment let be the maximum number of vertices of which are colorable with respect to . Define , where the minimum is taken over all list assignments with for all .
Conjecture [2] Let be a graph with list chromatic number and . Then
Conjecture If is the adjacency matrix of a -regular graph, then there is a symmetric signing of (i.e. replace some entries by ) so that the resulting matrix has all eigenvalues of magnitude at most .
Conjecture Let be an -uniform-partite hypergraph. If is the maximum number of pairwise disjoint edges in , and is the size of the smallest set of vertices which meets every edge, then .
Conjecture Let be the open unit disk in the complex plane and let be open sets such that . Suppose there are injective holomorphic functions such that for the differentials we have on any intersection . Then those differentials glue together to a meromorphic 1-form on .
Conjecture It has been shown that a -outerplanar embedding for which is minimal can be found in polynomial time. Does a similar result hold for -edge-outerplanar graphs?
Conjecture For every fixed and fixed colouring of with colours, there exists such that every colouring of the edges of contains either vertices whose edges are coloured according to or vertices whose edges are coloured with at most colours.
Conjecture Let be a graph and be a positive integer. The power of , denoted by , is defined on the vertex set , by connecting any two distinct vertices and with distance at most . In other words, . Also subdivision of , denoted by , is constructed by replacing each edge of with a path of length . Note that for , we have . Now we can define the fractional power of a graph as follows: Let be a graph and . The graph is defined by the power of the subdivision of . In other words . Conjecture. Let be a connected graph with and be a positive integer greater than 1. Then for any positive integer , we have . In [1], it was shown that this conjecture is true in some special cases.
Conjecture An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the vertices' trees, the star with vertices has the most endomorphisms, while the path with vertices has the least endomorphisms.
Conjecture For every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some constant .
Conjecture Can the approximation ratio be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than -hardness?
A -page book embedding of consists of a linear order of and a (non-proper) -colouring of such that edges with the same colour do not cross with respect to . That is, if for some edges , then and 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 , denoted by bt is the minimum integer for which there is a -page book embedding of .
Let be the graph obtained by subdividing each edge of exactly once.
Conjecture There is a function such that for every graph ,
Conjecture Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)
A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.
Question What is the Waring rank of the determinant of a generic matrix?
For simplicity say we work over the complex numbers. The generic matrix is the matrix with entries for . Its determinant is a homogeneous form of degree , in variables. If is a homogeneous form of degree , a power sum expression for is an expression of the form , the (homogeneous) linear forms. The Waring rank of is the least number of terms in any power sum expression for . For example, the expression means that has Waring rank (it can't be less than , as ).
The generic determinant (or ) has Waring rank . The Waring rank of the generic determinant is at least and no more than , see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's "VP versus VNP" problem.
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is a tournament of order 
, then it contains 
arc-disjoint transitive subtournaments of order 3. 
-spheres bound (rational) homology 
-balls?
is a directed graph with smallest directed cut of size 
, then 
in 
, let the symmetry group of 
ie: isotopy classes of diffeomorphisms of 
of 
is the unlink. Let 
be the subgroup of 
consisting of diffeomorphisms of 
given by restricting the diffeomorphism to the 
is a signed symmetric group -- the wreath product of a symmetric group with 
. 
so that every polytope of dimension 
has a 
let 
be the maximum number of vertices of 
, where the minimum is taken over all list assignments 
for all 
. 
and 
. Then ![\[\frac{\lambda_r}{r}\geq\frac{\lambda_s}{s}.\]](/files/tex/47be18e956355dd433b88b66eabf01a9e3ed5f61.png)
. The graph 
is defined to be the 
-power of the 
.
. Then 
. 
is the adjacency matrix of a 
entries by 
) so that the resulting matrix has all eigenvalues of magnitude at most 
. 
is no more than the maximal size of an induced 
be an 
-
is the maximum number of pairwise disjoint edges in 
is the size of the smallest set of vertices which meets every edge, then 
. 
. The chromatic number of its square is
if 
, \item at most 
if 
, \item at most 
if 
. 
contains every 2-regular graph on 
vertices as a minor. 
on 
from the edge set of 
, for all 
, and 
.
-regular graph 
for every 
with 
odd.
be an integer. If 
. 
be open sets such that 
. Suppose there are injective holomorphic functions 
such that for the differentials we have 
on any intersection 
. Then those differentials glue together to a meromorphic 1-form on 
? ![$ p \in {\mathbb Q}[x] $](/files/tex/a72fcb0a006c3c2afb3ba69722ccdb2599b83e90.png)
rationally derived if all roots of 
and the nonzero derivatives of 
. In other words, simplify this formula. 
denote the 
diagonal 
exists. 
denote the 
satisfies 
. 
and fixed colouring 
of 
with 
such that every colouring of the edges of 
contains either 
vertices whose edges are coloured with at most 
colours. 
power of 
, is defined on the vertex set 
, by connecting any two distinct vertices 
and 
with distance at most 
. Also 
, is constructed by replacing each edge 
of 
, we have 
.
power of the 
subdivision of 
.
and 
, we have 
.
Square free? 
of 
lines determined by 
. 
. 
be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than 
-hardness? 
of 
such that edges with the same colour do not cross with respect to 
for some edges 
, then 
and 
receive distinct colours. 
is the minimum integer 
be the graph obtained by subdividing each edge of 
such that for every graph 
and 
are monovalued, entirely defined funcoids with 
. Then there exists a pointfree funcoid 
such that (for every filter 
(The join operation is taken on the lattice of filters with reversed order.) 
generic matrix? 
for 
. Its determinant is a homogeneous form of degree 
variables. If 
is a homogeneous form of degree 
, the 
(homogeneous) linear forms. The Waring rank of 
means that 
(it can't be less than 
).
generic determinant 
(or 
) has Waring rank 
generic determinant is at least 
and no more than 
, see for instance 
be a set of ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
so that the graph induced by 
is acyclic for 
. 
-degenerate graph (a forest) and a 
-colourable. 
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