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 .
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.
Problem Let be natural numbers with . It follows from the pigeon-hole principle that there exist distinct subsets with . Is it possible to find such a pair in polynomial time?
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.
A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and . The circular flow number of is inf has a nowhere-zero -flow , and it is denoted by .
A graph with maximum vertex degree is a class 1 graph if its edge chromatic number is .
Conjecture Let be an integer and a -regular graph. If is a class 1 graph, then .
Conjecture Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.
Proposition It is easy to prove that is the infinitely small right neighborhood filter of point .
If proved true, the conjecture then can be generalized to a wider class of posets.
Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalent to the 4-sphere?
Conjecture Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.
Question Can either of the following be expressed in fixed-point logic plus counting: \item Given a graph, does it have a perfect matching, i.e., a set of edges such that every vertex is incident to exactly one edge from ? \item Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant?
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.
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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 
? 
be an elliptic curve over a number field 
. Then the order of the zeros of its 
-function, 
, at 
is the Mordell-Weil rank of 
. 
be a digraph all of whose strong components are nontrivial. Then 
. 
be the unique integer (with respect to a fixed 
) such that
is a prime iff ![$$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$](/files/tex/99af565f4cc4d3bab11eb3fbf54f78626678d484.png)
is a 
-edge-critical graph. Suppose that for each edge 
of 
of 
contains a directed path of length 
. 
be natural numbers with 
. It follows from the pigeon-hole principle that there exist distinct subsets 
with 
. Is it possible to find such a pair 
in polynomial time? 
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 
with the following property. If 
points in general position which is 2-colored, then it has 
monochromatic empty triangles. 
-flow 
on 
from the edge set of 
, for all 
, and 
. The circular flow number of 
has a nowhere-zero 
, and it is denoted by 
.
be an integer and 
-regular graph. If 
. 
be the complete funcoid corresponding to the usual topology on extended real line ![$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $](/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png)
. Let 
be the order on this set. Then 
is a complete funcoid. 
is the infinitely small right neighborhood filter of point ![$ x\in[-\infty,+\infty] $](/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png)
. 
be a matroid of rank 
let 
be the number of closed sets of rank 
.
is unimodal. 
is 
rectangular 
? 
if 
, \item at most 
if 
, \item at most 
if 
. 
. 
? 
covering design, or covering, is a family of 
-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 
.
. Find a procedure for constructing minimal coverings. 
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