Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of containing no pair of orthogonal vectors is attained by two open caps of geodesic radius around the north and south poles.
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?
Conjecture A Fermat prime is a Fermat number that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
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
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?
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.
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 .
Let be a graph. If and are two integers, a -colouring of is a function from to such that for each edge . Given a list assignment of , i.e.~a mapping that assigns to every vertex a set of non-negative integers, an -colouring of is a mapping such that for every . A list assignment is a --list-assignment if and for each vertex . Given such a list assignment , the graph G is --colourable if there exists a --colouring , i.e. is both a -colouring and an -colouring. For any real number , the graph is --choosable if it is --colourable for every --list-assignment . Last, is circularly -choosable if it is --choosable for any , . The circular choosability (or circular list chromatic number or circular choice number) of G is
Problem What is the best upper bound on circular choosability for planar graphs?
A covering design, or covering, is a family of -subsets, called blocks, chosen from a -set, such that each -subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by .
Problem Find a closed form, recurrence, or better bounds for . Find a procedure for constructing minimal coverings.
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.
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).
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containing no pair of orthogonal vectors is attained by two open caps of geodesic radius 
around the north and south poles. 
be a 
-dimensional smooth submanifold of 
, 
. By the Jordan-Brouwer separation theorem, 
-manifolds which share 
? ie: is ![\[ F_n = 2^{2^n } + 1 \]](/files/tex/0da5a50010e4e5df91c0d58080245ece34ec9ca6.png)
that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
can be replaced by a smaller value, we can also write the conjecture as, for any positive 
be a planar graph of maximum degree 
. The chromatic number of its square is
if 
, \item at most 
if 
, \item at most 
if 
. 
vertices has an edge colouring such that:
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)
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.
containing a Hamiltonian path.
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 
. 
be a sequence of nonnegative integers which strictly decreases until 
.
with 
crossings, but not with less crossings. 
is the adjacency matrix of a 
-regular graph, then there is a symmetric signing of 
entries by 
) so that the resulting matrix has all eigenvalues of magnitude at most 
. 
be abelian groups and let 
and 
satisfy 
and 
. If there is a 
to 
, then every graph with a B-flow has a B'-flow. 
denote the 
diagonal 
exists. 
is a prime iff 
rectangular 
inside an 
? 
be a graph. If 
are two integers, a 
-colouring of 
from 
to 
such that 
for each edge 
. Given a list assignment 
of 
a set of non-negative integers, an 
such that 
for every 
. A list assignment 
-
and 
for each vertex 
. Given such a list assignment 
, the graph 
covering design, or covering, is a family of 
.
. Find a procedure for constructing minimal coverings. 
. The representation function 
for 
is given by the rule 
. We call 
is never 
, 
does not divide 
where 
is the Legendre symbol. 
. 
is arbitrarily large? 
for every endo-
-graph is an 
for every 
with odd size.
for every 
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). 
. 
in the hypercube. ![\[ F_n = 2^{2^{n } } + 1 \]](/files/tex/70ca73d7e82af2fee084a8417e172c58cf78b376.png)
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