Question I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all . Suppose X is an m-by-n integer matrix . Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
Conjecture For , let be the statement that given any exact -coloring of the edges of a complete countably infinite graph (that is, a coloring with colors all of which must be used at least once), there exists an exactly -colored countably infinite complete subgraph. Then is true if and only if , , or .
Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an grid. The first player (if any) to occupy a set of cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play?
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 If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.
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 Does there exist a polynomial time algorithm which takes as input a graph and for every vertex a subset of , and decides if there exists a partition of into so that only if and so that are independent, is a clique, and there are no edges between and ?
Conjecture Let be a -connected cubic graph and let be a -regular subgraph such that is connected. Then has a cycle double cover which contains (i.e all cycles of ).
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 Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?
Conjecture If is a bridgelesscubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .
Conjecture Let be a field of characteristic zero. A collection of polynomials in variables defines an automorphism of if and only if the Jacobian matrix is a nonzero constant.
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.
Problem Let be positve integer Does there exists an integer such that every -strong tournament admits a partition of its vertex set such that the subtournament induced by is a non-trivial -strong for all .
If is a finite set of points which is 2-colored, an empty triangle is a set with so that the convex hull of is disjoint from . We say that is monochromatic if all points in are the same color.
Conjecture There exists a fixed constant with the following property. If is a set of points in general position which is 2-colored, then it has monochromatic empty triangles.
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.
An oriented colouring of an oriented graph is assignment of colours to the vertices such that no two arcs receive ordered pairs of colours and . It is equivalent to a homomorphism of the digraph onto some tournament of order .
Conjecture Denote by the number of non-Hamiltonian 3-regular graphs of size , and similarly denote by the number of non-Hamiltonian 3-regular 1-connected graphs of size .
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are independent random variables with ![$ \mathbb{E}[X_i] \leq \mu $](/files/tex/e0268221532981debea25e9446c8ee6f112e1881.png)
, then ![$$\mathrm{Pr} \left( \sum X_i - \mathbb{E} \left[ \sum X_i \right ] < \delta \mu \right) \geq \min \left ( (1 + \delta)^{-1} \delta, e^{-1} \right).$$](/files/tex/03dc1130142ee6fefcc33888e2fb6137211bf327.png)
has chromatic number at most 
.
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 
-graph if it has a bipartition 
so that every vertex in 
has degree 
and every vertex in 
has degree 
. 
. Suppose X is an m-by-n integer matrix 
. Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
, let 
be the statement that given any exact 
-coloring of the edges of a complete countably infinite graph (that is, a coloring with 
-colored countably infinite complete subgraph. Then 
, 
, or 
. 
grid. The first player (if any) to occupy a set of 
cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play? 
-size circuits compute the function 
on 
defined inductively by 
, 
, 
, and 
? 
be a graph and 
be a positive integer. The 
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 
.
. The graph 
is defined by the 
power of the 
subdivision of 
.
and 
, we have 
.
contains a complete bipartite subgraph with bipartition 
? 
be a matroid of rank 
, and for 
let 
be the number of closed sets of rank 
.
is unimodal. 
, whether there exists a homomorphism from 
for some constant 
be a set, 
be the set of filters on 
be the set of principal filters on 
.
, then 
is a completary multifuncoid of the form 
. 
of 
a subset 
of 
, and decides if there exists a partition of 
so that 
only if 
and so that 
are independent, 
is a clique, and there are no edges between 
and 
? 
-connected cubic graph and let 
be a 
is connected. Then 
containing no pair of orthogonal vectors is attained by two open caps of geodesic radius 
around the north and south poles. 
be abelian groups and let 
and 
satisfy 
and 
. If there is a 
to 
, then every graph with a B-flow has a B'-flow. 
which cannot be expressed as a union of 
, that every sufficiently large 
vertices whose deletion results in a 
.
-connected planar graph, then 
has a Hamilton cycle. 
of 
there is an integer 
such that every set of at least 
collinear points or an empty hexagon. 
on an even number of vertices can be decomposed into 
directed paths. 
which is slice, is it a ribbon knot?
of polynomials in variables 
defines an automorphism of 
if and only if the Jacobian matrix is a nonzero constant. 
is a tournament of order 
arc-disjoint transitive subtournaments of order 3. 
is a connected cubic graph admitting a 
such that the cubic graph homeomorphic to 
has a 
. 
in 
such that the fundamental group of the complement has an unsolvable word problem? 
. Is it true that for large enough~
? 
denote ''
divides 
''. The mimic function in number theory is defined as follows [1].
divisible by 
, the mimic function, 
, is given by,
. 
is prime and 
, we write 
if 
satisfies 
. The problem of finding such an integer 
(with 
) is the Discrete Log Problem.
be positve integer Does there exists an integer 
such that every 
of its vertex set such that the subtournament induced by 
is a non-trivial 
-strong for all 
. 
. Does the fragment 
have the finite model property? 
let 
. 
, let 
be a map, put 
and 
. Then 
is a finite set of points which is 2-colored, an empty triangle is a set 
with 
so that the convex hull of 
. We say that 
monochromatic empty triangles. 
so that every polytope of dimension 
has a 
and 
. It is equivalent to a homomorphism of the digraph onto some tournament of order 
the number of non-Hamiltonian 3-regular graphs of size 
, and similarly denote by 
the number of non-Hamiltonian 3-regular 1-connected graphs of size 
? 
has a homomorphism to 
. 
-
of 
so that 
is 
is 
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