Problem What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?
Conjecture If in a bridgeless cubic graph the cycles of any -factor are odd, then , where denotes the oddness of the graph , that is, the minimum number of odd cycles in a -factor of .
Given integers , the 2-stage Shuffle-Exchange graph/network, denoted , is the simple -regular bipartite graph with the ordered pair of linearly labeled parts and , where , such that vertices and are adjacent if and only if (see Fig.1).
Given integers , the -stage Shuffle-Exchange graph/network, denoted , is the proper (i.e., respecting all the orders) concatenation of identical copies of (see Fig.1).
Let be the smallest integer such that the graph is rearrangeable.
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.
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
Conjecture Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)
A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.
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 Let be the open unit disk in the complex plane and let be open sets such that . Suppose there are injective holomorphic functions such that for the differentials we have on any intersection . Then those differentials glue together to a meromorphic 1-form on .
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.
Question What is the Waring rank of the determinant of a generic matrix?
For simplicity say we work over the complex numbers. The generic matrix is the matrix with entries for . Its determinant is a homogeneous form of degree , in variables. If is a homogeneous form of degree , a power sum expression for is an expression of the form , the (homogeneous) linear forms. The Waring rank of is the least number of terms in any power sum expression for . For example, the expression means that has Waring rank (it can't be less than , as ).
The generic determinant (or ) has Waring rank . The Waring rank of the generic determinant is at least and no more than , see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's "VP versus VNP" problem.
Conjecture For every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some constant .
Conjecture Let is a -separable (the same as for symmetric transitive) compact funcoid and is a uniform space (reflexive, symmetric, and transitive endoreloid) such that . Then .
The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.
The direct proof may be constructed by correcting all errors an omissions in this draft article.
Direct proof could be better because with it we would get a little more general statement like this:
Conjecture Let be a -separable compact reflexive symmetric funcoid and be a reloid such that \item ; \item .
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.
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the cycles of any 
-factor are odd, then 
, where 
denotes the oddness of the graph 
for principal funcoid 
and a set 
of funcoids of appropriate sources and destinations. 
. 
, the 2-stage Shuffle-Exchange graph/network, denoted 
, is the simple 
-regular bipartite graph with the ordered pair 
of linearly labeled parts 
and 
, where 
, such that vertices 
and 
are adjacent if and only if 
(see Fig.1).
, the 
-stage Shuffle-Exchange graph/network, denoted 
, is the proper (i.e., respecting all the orders) concatenation of 
identical copies of 
be the smallest integer 
such that the graph 
. 
be an abelian group. Is the 
with 
so that 
has a 
. 
-graph if it has a bipartition 
so that every vertex in 
has degree 
and every vertex in 
has degree 
. 
matrix, then there is an ![$ [A A] $](/files/tex/d1e9d82c656535b507686183e640178057fae455.png)
so that 
is nonzero. 
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 
.
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)
, whether there exists a homomorphism from 
for some constant 
? 
and 
are monovalued, entirely defined funcoids with 
. Then there exists a pointfree funcoid 
such that (for every filter 
(The join operation is taken on the lattice of filters with reversed order.) 
be the open unit disk in the complex plane and let 
be open sets such that 
. Suppose there are injective holomorphic functions 
such that for the differentials we have 
on any intersection 
. Then those differentials glue together to a meromorphic 1-form on 
-edge-critical graph. Suppose that for each edge 
of 
of 
-edge-colorable unless all lists are equal to each other. 
vertices, then 
. 
, then 
generic matrix? 
for 
. Its determinant is a homogeneous form of degree 
, in 
variables. If 
is a homogeneous form of degree 
, the 
(homogeneous) linear forms. The Waring rank of 
means that 
has Waring rank 
).
generic determinant 
(or 
) has Waring rank 
. The Waring rank of the 
generic determinant is at least 
and no more than 
, see for instance 
of 
points in the plane, not all collinear, there is a point in 
lines determined by 
-separable (the same as 
for symmetric transitive) compact funcoid and 
is a uniform space (reflexive, symmetric, and transitive endoreloid) such that 
. Then 
. 
. 
. 
vertices. 
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