Conjecture There exists an integer such that every -arc-strong digraph with specified vertices and contains an out-branching rooted at and an in-branching rooted at which are arc-disjoint.
To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.
Remark: It appears maximizing the total perimeter is the easier problem.
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.
Conjecture Let if is odd and if is even. Let . Assume we start with some number and repeatedly take the of the current number. Prove that no matter what the initial number is we eventually reach .
Conjecture Suppose that is a -edge-critical graph. Suppose that for each edge of , there is a list of colors. Then is -edge-colorable unless all lists are equal to each other.
We are given a complete simple undirected weighted graph and its first arbitrary shortest spanning tree . We define the next graph and find on the second arbitrary shortest spanning tree . We continue similarly by finding on , etc. Let k be the smallest number of disjoint shortest spanning trees as defined above and let be the graph obtained as union of all disjoint trees.
Question 1. What is the smallest number of disjoint spanning trees creates a graph containing a Hamiltonian path.
Question 2. What is the smallest number of disjoint spanning trees creates a graph containing a shortest Hamiltonian path?
Questions 3 and 4. Replace in questions 1 and 2 a shortest spanning tree by a 1-tree. What is the smallest number of disjoint 1-trees creates a Hamiltonian graph? What is the smallest number of disjoint 1-trees creates a graph containing a shortest Hamiltonian cycle?
Conjecture Suppose with is a connected cubic graph admitting a -edge coloring. Then there is an edge such that the cubic graph homeomorphic to has a -edge coloring.
Question Can either of the following be expressed in fixed-point logic plus counting: \item Given a graph, does it have a perfect matching, i.e., a set of edges such that every vertex is incident to exactly one edge from ? \item Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant?
For a graph , let denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let denote the cardinality of a minimum feedback vertex set (set of vertices so that is acyclic).
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 If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.
Conjecture Let and . Then for any neighborhood there is such that is periodic point of
There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for . In the case of Flows this has been solved by Sushei Hayahshy for . But in the two cases the problem is wide open for
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.
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such that every 
with specified vertices 
and 
contains an out-branching rooted at 
be an indexed family of sets.
for 
.
on the lattice 
of all finite unions of products.
a bijection from hyperfuncoids 
defining the inverse bijection? If not, characterize the image of the function 
of complements of elements of the set 
without a 
.
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. 
if 
is odd and 
if 
. Assume we start with some number 
of the current number. Prove that no matter what the initial number is we eventually reach 
. 
is 
we have 
. Then 
is graphic. 
-edge-critical graph. Suppose that for each edge 
of 
of 
-edge-colorable unless all lists are equal to each other. 
and its first arbitrary shortest spanning tree 
. We define the next graph 
and find on 
the second arbitrary shortest spanning tree 
. We continue similarly by finding 
on 
, etc. Let k be the smallest number of disjoint shortest spanning trees as defined above and let 
be the graph obtained as union of all 
containing a Hamiltonian path.
-connected finite graph, is there an assignment of lengths 
to the edges of 
-geodesic cycle is 
is a connected cubic graph admitting a 
such that the cubic graph homeomorphic to 
has a 
, there is an integer 
so that every strongly 
, every (loopless) graph 
immerses 
. 
of edges such that every vertex is incident to exactly one edge from 
denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let 
denote the cardinality of a minimum feedback vertex set (set of vertices 
so that 
is acyclic). 
. 
structurally stable diffeomorphism is hyperbolic. 
is an invertible 
matrix, then there is an 
of ![$ [A A] $](/files/tex/d1e9d82c656535b507686183e640178057fae455.png)
so that 
is nonzero. 
, 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 
. 
-graph if it has a bipartition 
so that every vertex in 
and every vertex in 
. 
-
is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. 
and 
. Then for any neighborhood 
there is 
such that 
is periodic point of 
vector fields . In the case of diffeos this was proved by Charles Pugh for 
. In the case of Flows this has been solved by Sushei Hayahshy for 
is the only 
be a set, 
be the set of filters on 
be the set of principal filters on 
.
, then 
is a completary multifuncoid of the form 
. 
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