Problem The valency-variety of a graph is the number of different degrees in . Is the chromatic number of any graph with at least two vertices greater than
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 Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
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 It has been shown that a -outerplanar embedding for which is minimal can be found in polynomial time. Does a similar result hold for -edge-outerplanar graphs?
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.
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.
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 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.
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.
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 an integer. For every integer , there exists an integer such that for every digraph , either has a pairwise-disjoint directed cycles of length at least , or there exists a set of at most vertices such that has no directed cycles of length at least .
Problem () Find a sufficient condition for a straight -stage graph to be rearrangeable. In particular, what about a straight uniform graph?
Conjecture () Let be a simple regular ordered -stage graph. Suppose that the graph is externally connected, for some . Then the graph is rearrangeable.
Problem (2) Find a composite or which divides both (see Fermat pseudoprime) and the Fibonacci number (see Lucas pseudoprime), or prove that there is no such .
Problem What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or antiprism?
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 ).
Problem Let be an indexed family of filters on sets. Which of the below items are always pairwise equal?
1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters .
2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters .
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.
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 .
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be an integer and let 
denote the least integer 
such that there exists a simple graph on 
spanning trees. Then 
of a graph 
is the number of different degrees in 
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 ![\[ M_n = 2^p - 1 \]](/files/tex/eb18a56e5c2e8b1be6ac733d217c0c1f1a47a94e.png)
that is prime.
there is 
such that, for every large 
such that 
and 
. 
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
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.
be a set of points in the plane. Two points 
and 
in 
there is an integer 
collinear points or 
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 
. 
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. 
covering design, or covering, is a family of 
-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 
.
. Find a procedure for constructing minimal coverings. 
grid. The first player (if any) to occupy a set of 
be an integer. For every integer 
, there exists an integer 
such that for every digraph 
of at most 
vertices such that 
has no directed cycles of length at least 
) Find a sufficient condition for a straight 
) Let 
be a simple regular ordered 
-stage graph. Suppose that the graph 
is externally connected, for some 
. Then the graph 
is rearrangeable. 
has a proper 
-
or 
which divides both 
(see 
(see 
. 
is 
? 
is connected. Then 
be an indexed family of filters on sets. Which of the below items are always pairwise equal?
. 
of 
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CSI of Charles University