Conjecture Is it possible to color edges of the complete graph using colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
Equivalently: is the star chromatic index of linear in ?
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 .
Let be a set, be the set of filters on ordered reverse to set-theoretic inclusion, be the set of principal filters on , let be an index set. Consider the filtrator .
Conjecture If is a completary multifuncoid of the form , then is a completary multifuncoid of the form .
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.
Problem Given two codes , their Tensor Product is the code that consists of the matrices whose rows are codewords of and whose columns are codewords of . The product is said to be robust if whenever a matrix is far from , the rows (columns) of are far from (, respectively).
The problem is to give a characterization of the pairs whose tensor product is robust.
We say that a set is -universal if every vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in , and all edges are (non-intersecting) straight line segments.
Question Does there exist an -universal set of size ?
Conjecture Given any complex numbers which are linearly independent over the rational numbers , then the extension field has transcendence degree of at least over .
Conjecture For every graph without a bridge, there is a flow .
Conjecture There exists a map so that antipodal points of receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.
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.
Conjecture If is the maximal degree of a graph , then is strongly -colorable.
Conjecture Is the approximation ratio for the Maximum Edge Disjoint Paths (MaxEDP) or the Maximum Integer Multiflow problem (MaxIMF) bounded by a constant in -outerplanar graphs or tree-width graphs?
Conjecture If is a cubic graph not containing a triangle, then it is possible to color the edges of by five colors, so that the complement of every color class is a bipartite graph.
Conjecture There exists a fixed constant so that every abelian group has a subset with so that the Cayley graph has no clique or independent set of size .
Conjecture Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.
Proposition It is easy to prove that is the infinitely small right neighborhood filter of point .
If proved true, the conjecture then can be generalized to a wider class of posets.
Problem Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?
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be a perfect graph on 
vertices. Is it true that either 
contains a complete bipartite subgraph with bipartition 
so that 
? 
using 
colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
? 
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 
. 
be a set, 
be the set of filters on 
be the set of principal filters on 
.
is a completary multifuncoid of the form 
, then 
is a completary multifuncoid of the form 
. 
so that all pairwise distances between points in 
are rational? 
, their Tensor Product 
is the code that consists of the matrices whose rows are codewords of 
and whose columns are codewords of 
. The product 
is far from 
-flow 
on 
of 
from the edge set of 
, for all 
, and 
.
-regular graph 
for every 
with 
odd.
be an integer. If 
. 
is 
, and all edges are (non-intersecting) straight line segments. 
which are linearly independent over the rational numbers 
, then the extension field 
has transcendence degree of at least 
, every digraph with minimum out-degree at least 
contains 
into 
so that the graph induced by 
is acyclic for 
. 
.
so that antipodal points of 
receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero. 
is the maximal degree of a graph 
-colorable. 
, the 
, is defined as the sum of all 
by using partitions. 
is a tournament of order 
arc-disjoint transitive subtournaments of order 3. 
people admits a solution is 
. 
so that every abelian group 
with 
so that the 
has no clique or independent set of size 
. ![$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $](/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png)
. Let 
be the order on this set. Then 
is a complete funcoid. 
is the infinitely small right neighborhood filter of point ![$ x\in[-\infty,+\infty] $](/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png)
. 
be the flow polynomial of a graph 
equals the number of 
for every 2-edge-connected graph 
-dimensional smooth submanifold of 
, 
-manifolds which share 
? ie: is 
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