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   Every planar oriented graph has an acyclic induced subdigraph of order at least .

Conjecture   Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.

Conjecture   Twin Prime Conjecture: There are an infinite number of twin primes.

Conjecture   A natural number is a prime iff

Conjecture   Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a deterministic Turing machine in polynomial time? That is, does P = BPP?

Conjecture   Every even integer greater than 2 is the sum of two primes.

Conjecture   Every graph satisfies .

Conjecture   Let be a digraph all of whose strong components are nontrivial. Then contains a cyclic spanning subdigraph with cyclomatic number at most .

Question   I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all . Suppose X is an m-by-n integer matrix . Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.

Conjecture   If is a tournament of order , then it contains arc-disjoint transitive subtournaments of order 3.

Conjecture   If is a -regular directed graph with no parallel arcs, then contains a collection of arc-disjoint directed cycles.

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

Conjecture   Let and . Then for any neighborhood there is such that is periodic point of

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.

Question   Are there graphs for which is arbitrarily large?

Problem   Are almost all graphs uniquely determined by the spectrum of their adjacency matrix?

Conjecture   If is the adjacency matrix of a -regular graph, then there is a symmetric signing of (i.e. replace some entries by ) so that the resulting matrix has all eigenvalues of magnitude at most .

Conjecture  (BEC-conjecture)   If and are -vertex graphs and , then and pack.

Conjecture   Every k-arc-strong tournament decomposes into k spanning strong digraphs.

Question   Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments?