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 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 In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical product:
\item Product morphism is defined similarly to the category of topological spaces. \item Product object is the sub-atomic product. \item Projections are sub-atomic projections.
See details, exact definitions, and attempted proofs here.
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.
Given a finite family of graphs and an integer , the Turán number of is the largest integer such that there exists a graph on vertices with edges which contains no member of as a subgraph.
Conjecture For every finite family of graphs there exists an such that .
Conjecture Given any complex numbers which are linearly independent over the rational numbers , then the extension field has transcendence degree of at least over .
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.
Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.
Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and same perimeter?
Definitions: Define a Fair Partition of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a Convex Fair Partition.
Questions: 1. (Rephrasing the above 'basic' question) Given any positive integer n, can any convex polygon be convex fair partitioned into n pieces?
2. If the answer to the above is "Not always'', how does one decide the possibility of such a partition for a given convex polygon and a given n? And if fair convex partition is allowed by a specific convex polygon for a give n, how does one find the optimal convex fair partition that minimizes the total length of the cut segments?
3. Finally, what could one say about higher dimensional analogs of this question?
Conjecture: The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: Every convex polygon allows a convex fair partition into n pieces for any n
The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the following atomic formulas:
\item \item
where is a fixed recursive set of integers.
Let us fix and a closed formula in this language.
Conjecture Is it true that the validity of for a graph of tree-width at most can be tested in polynomial time in the size of ?
The deck of a graph is the multiset consisting of all unlabelled subgraphs obtained from by deleting a vertex in all possible ways (counted according to multiplicity).
Conjecture If two graphs on vertices have the same deck, then they are isomorphic.
A -page book embedding of consists of a linear order of and a (non-proper) -colouring of such that edges with the same colour do not cross with respect to . That is, if for some edges , then and receive distinct colours.
One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.
The book thickness of , denoted by bt is the minimum integer for which there is a -page book embedding of .
Let be the graph obtained by subdividing each edge of exactly once.
Conjecture There is a function such that for every graph ,
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with no 
with 
so that 
has a 
. 
, there exists a Fibonacci number divisible by 
, 
does not divide 
where 
is the Legendre symbol. 
of a locally finite graph 
-connected graph with a Hamilton cycle has a second Hamilton cycle. 
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.) 
-
of 
so that 
is 
-edge-connected and 
is 
-edge-connected? 
be abelian groups and let 
and 
satisfy 
and 
. If there is a 
to 
, then every graph with a B-flow has a B'-flow. 
-edge-critical graph. Suppose that for each edge 
of 
of 
-edge-colorable unless all lists are equal to each other. 
the following inequality holds 
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 
in any power sum expression for 
means that 
has Waring rank 
(it can't be less than 
).
generic determinant 
(or 
) has Waring rank 
generic determinant is at least 
and no more than 
, see for instance 
of graphs and an integer 
, the Turán number 
of 
such that there exists a graph on 
such that 
.
which are linearly independent over the rational numbers 
, then the extension field 
has transcendence degree of at least 
. The graph 
is defined to be the 
.
. Then 
. 
\item 
and a closed formula 
denote the 
satisfies 
. 
and 
vertices have the same deck, then they are isomorphic. 
-connected finite graph, is there an assignment of lengths 
to the edges of 
-geodesic cycle is 
of 
and a (non-proper) 
for some edges 
, then 
and 
is the minimum integer 
be the graph obtained by subdividing each edge of 
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