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.