For a graph , let denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let denote the cardinality of a minimum feedback vertex set (set of vertices so that is acyclic).
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 the space of Diffeomorphisms on the connected , compact and boundaryles manifold M and the space of vector fields. There is a dense set ( ) such that exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space
This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .
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.
Conjecture If is a simple graph which is the union of pairwise edge-disjoint complete graphs, each of which has vertices, then the chromatic number of is .
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
Problem Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?
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.
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 .
Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number .
The number appears once in Pascal's triangle, appears twice, appears three times, and appears times. There are infinite families of numbers known to appear times. The only number known to appear times is . It is not known whether any number appears more than times. The conjectured upper bound could be ; Singmaster thought it might be or . See Singmaster's conjecture.
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, let 
denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let 
denote the cardinality of a minimum feedback vertex set (set of vertices 
so that 
is acyclic). 
. 
then 
for every funcoid 
and atomic f.o. 
and 
on the source and destination of 
then 
for every funcoid 
, 
. 
denote the 
-dimensional cube graph. A map 
is called edge-antipodal if 
whenever 
are antipodal edges. 
and 
which are joined by a monochromatic path. 
is a sequence of elements from a multiplicative group of order 
, then there exist 
so that 
. 
such that for every hexagonal graph 
, 
such that for any pair 
there are 
edge-disjoint paths from 
to 
in 
edge-disjoint trees, each of which contains 
. 
-minor-free graph is 
-list-colourable for some constant 
. 
be the space of 
Diffeomorphisms on the connected , compact and boundaryles manifold M and 
the space of 
(
) such that 
exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space 
vertices, then 
. 
. 
, there exists a Fibonacci number divisible by 
, 
does not divide 
where 
is the Legendre symbol. 
vertices have the same deck, then they are isomorphic. 
, every (loopless) graph 
immerses 
is a 2-transition system for 
has a compaible decomposition. 
such that 
or prove that such integers do not exist. 
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)
, that every sufficiently large 
vertices whose deletion results in a 
contains a complete bipartite subgraph with bipartition 
so that 
? 
-
of 
so that 
is 
is 
. 
-edge-critical graph. Suppose that for each edge 
of 
of 
-edge-colorable unless all lists are equal to each other. 
be an integer. For every integer 
, there exists an integer 
such that for every digraph 
, either 
, or there exists a set 
vertices such that 
has no directed cycles of length at least 
. 
appears once in Pascal's triangle, 
appears twice, 
appears three times, and 
appears 
times. There are infinite families of numbers known to appear 
times is 
. It is not known whether any number appears more than 
. See 
to the edges of 
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