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.
Suppose is a finite group, and is a positive integer dividing . Suppose that has exactly solutions to . Does it follow that these solutions form a subgroup of ?
Conjecture The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the following: 2, 4, 6, 7, 14, 3^k, k >=1.
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 \item If is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. \item If the line graph of a locally finite graph is 4-connected, then is hamiltonian.
Conjecture Let be the space of Diffeomorphisms on the connected , compact and boundaryles manifold M and the space of vector fields. There is a dense set ( ) such that exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space
This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .
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.
Conjecture If is a finite field with at least 4 elements and is an invertible matrix with entries in , then there are column vectors which have no coordinates equal to zero such that .
Let be a set of points in the plane. Two points and in are visible with respect to if the line segment between and contains no other point in .
Conjecture For all integers there is an integer such that every set of at least points in the plane contains at least collinear points or pairwise visible points.
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 .
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.
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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 
be a positive integer greater than 1. Then for any positive integer 
, we have 
.
. Is it true that for large enough~
? 
is a positive integer dividing 
. Suppose that 
. Does it follow that these solutions form a subgroup of 
-uniform hypergraph on 
vertices which contains no complete 
hyperedges. 
vertices which contains no complete 
hyperedges. 
concentrated on a single value? 
of a locally finite graph 
be the space of 
Diffeomorphisms on the connected , compact and boundaryles manifold M and 
the space of 
(
) such that 
exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space 
containing no pair of orthogonal vectors is attained by two open caps of geodesic radius 
around the north and south poles. 
is a finite field with at least 4 elements and 
is an invertible 
matrix with entries in 
which have no coordinates equal to zero such that 
. 
such that every 
-minor-free graph has at most 
cliques? 
be a set of points in the plane. Two points 
and 
in 
there is an integer 
collinear points or 
, it is impossible to cover a square of side greater than 
unit squares. 
is the minimum number 
for which there exists a polytope 
with 
with 
.
? 
we have 
. Then 
is graphic. 
the following inequality holds 
has distinct subset sums if distinct subsets of 
whenever 
has distinct subset sums. ![$ p \in {\mathbb Q}[x] $](/files/tex/a72fcb0a006c3c2afb3ba69722ccdb2599b83e90.png)
rationally derived if all roots of 
and the nonzero derivatives of 
arc-disjoint directed cycles. 
. 
of 
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 
monochromatic empty triangles. 
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