Conjecture Let be a cubic graph with no bridge. Then there is a coloring of the edges of using the edges of the Petersen graph so that any three mutually adjacent edges of map to three mutually adjancent edges in the Petersen graph.
Let be a hypergraph. A strongly maximal matching is a matching so that for every matching . A strongly minimal cover is a (vertex) cover so that for every cover .
Conjecture If is a (possibly infinite) hypergraph in which all edges have size for some integer , then has a strongly maximal matching and a strongly minimal cover.
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 Does there exist a polynomial time algorithm which takes as input a graph and for every vertex a subset of , and decides if there exists a partition of into so that only if and so that are independent, is a clique, and there are no edges between and ?
Conjecture The following statements are equivalent for every endofuncoid and a set : \item is connected regarding . \item For every there exists a totally ordered set such that , , and for every partion of into two sets , such that , we have .
Conjecture Every surreal number has a unique sign expansion, i.e. function , where is some ordinal. This is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of as .
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.
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 friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.
Problem Is it true that for every , all but finitely many -regular graphs have friendly partitions?
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.
Conjecture For every fixed graph , there exists a constant , so that every graph without an induced subgraph isomorphic to contains either a clique or an independent set of size .
Conjecture Let be a graph and let such that for any pair there are edge-disjoint paths from to in . Then contains edge-disjoint trees, each of which contains .
Conjecture Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
Problem Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?
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be a 
and graph 
. 
so that 
for every matching 
. A strongly minimal cover is a (vertex) cover 
so that 
for every cover 
. 
for some integer 
, then 
be an integer. For every integer 
, there exists an integer 
such that for every digraph 
, either 
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 
. 
vertices, then 
. 
is defined to be 
where 
is the number of faces of 
.
so that every convex 4-polytope has fatness at most 
a subset 
of 
, and decides if there exists a partition of 
into 
so that 
only if 
and so that 
are independent, 
is a clique, and there are no edges between 
and 
? 
be a 
is connected regarding 
.
and a set 
:
there exists a totally ordered set 
such that 
, 
, and for every partion 
of 
, 
such that 
, we have ![$ X \mathrel{[ \mu]^{\ast}} Y $](/files/tex/0ef560be389646efd1fdde5ebc9afc9ac98ee64e.png)
. 
, where 
is some ordinal. This 
as 
.
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 
. The Waring rank of the 
generic determinant is at least 
and no more than 
, see for instance 
of edges such that every vertex is incident to exactly one edge from 
. 
so that the power series 
is bounded for all 
. 
. The graph 
is defined to be the 
-power of the 
.
. Then 
. 
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. 
-
of 
so that 
is 
-edge-connected and 
is 
-edge-connected? 
denote the 
? 
be integers. Set 
and 
for 
. Eventually we have 
; put 
.
, since 
, 
, 
, 
, 
, 
, 
, 
.
.
, so that every graph 
. 
is at most 
.
such that for any pair 
there are 
edge-disjoint paths from 
to 
. The representation function 
for 
is given by the rule 
. We call 
is never 
.
and 
are 
, then 
for principal funcoid 
and a set 
of funcoids of appropriate sources and destinations. 
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)
-dimensional smooth submanifold of 
, 
. By the Jordan-Brouwer separation theorem, 
? ie: is 
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