# Recent Activity

## Are all Fermat Numbers square-free? ★★★

Author(s):

Conjecture   Are all Fermat Numbers Square-Free?

Keywords:

## Hedetniemi's Conjecture ★★★

Author(s): Hedetniemi

Conjecture   If are simple finite graphs, then .

Here is the tensor product (also called the direct or categorical product) of and .

## Choosability of Graph Powers ★★

Author(s): Noel

Question  (Noel, 2013)   Does there exist a function such that for every graph ,

## Large acyclic induced subdigraph in a planar oriented graph. ★★

Author(s): Harutyunyan

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

Keywords:

## Polignac's Conjecture ★★★

Author(s): de Polignac

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.

In particular, this implies:

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

Keywords: prime; prime gap

## Alexa's Conjecture on Primality ★★

Author(s): Alexa

Definition   Let be the unique integer (with respect to a fixed ) such that

Conjecture   A natural number is a prime iff

Keywords: primality

## P vs. BPP ★★★

Author(s): Folklore

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?

Keywords: BPP; circuit complexity; pseudorandom generators

## Goldbach conjecture ★★★★

Author(s): Goldbach

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

Keywords: additive basis; prime

## Goldberg's conjecture ★★★

Author(s): Goldberg

The overfull parameter is defined as follows:

Conjecture   Every graph satisfies .

Keywords: edge-coloring; multigraph

## Cyclic spanning subdigraph with small cyclomatic number ★★

Author(s): Bondy

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

Keywords:

## inverse of an integer matrix ★★

Author(s): Gregory

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.

## Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament ★★

Author(s): Yuster

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

Keywords:

## Arc-disjoint directed cycles in regular directed graphs ★★

Author(s): Alon; McDiarmid; Molloy

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

Keywords:

## Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems) ★★★★

Author(s):

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 .

Keywords: Attractors , basins, Finite

## Closing Lemma for Diffeomorphism (Dynamical Systems) ★★★★

Author(s): Charles Pugh

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

There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for . In the case of Flows this has been solved by Sushei Hayahshy for . But in the two cases the problem is wide open for

Keywords: Dynamics , Pertubation

## Sub-atomic product of funcoids is a categorical product ★★

Author(s):

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.

See details, exact definitions, and attempted proofs here.

Keywords:

## Bounding the on-line choice number in terms of the choice number ★★

Author(s): Zhu

Question   Are there graphs for which is arbitrarily large?

## Are almost all graphs determined by their spectrum? ★★★

Author(s):

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

Keywords: cospectral; graph invariant; spectrum

## Signing a graph to have small magnitude eigenvalues ★★

Author(s): Bilu; Linial

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 .

Keywords: eigenvalue; expander; Ramanujan graph; signed graph; signing

## The Bollobás-Eldridge-Catlin Conjecture on graph packing ★★★

Author(s):

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

Keywords: graph packing