# Recent Activity

## List Total Colouring Conjecture ★★

Author(s): Borodin; Kostochka; Woodall

Conjecture   If is the total graph of a multigraph, then .

Keywords: list coloring; Total coloring; total graphs

## Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by projective plane we mean the set of all lines through the origin in .

Definition   Say that a subset of the projective plane is octahedral if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition   Say that a subset of the projective plane is weakly octahedral if every set such that is octahedral.
Conjecture   Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.

Keywords: Partitioning; projective plane

## Kriesell's Conjecture ★★

Author(s): Kriesell

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 .

## 2-colouring a graph without a monochromatic maximum clique ★★

Author(s): Hoang; McDiarmid

Conjecture   If is a non-empty graph containing no induced odd cycle of length at least , then there is a -vertex colouring of in which no maximum clique is monochromatic.

Keywords: maximum clique; Partitioning

## Almost all non-Hamiltonian 3-regular graphs are 1-connected ★★

Author(s): Haythorpe

Conjecture   Denote by the number of non-Hamiltonian 3-regular graphs of size , and similarly denote by the number of non-Hamiltonian 3-regular 1-connected graphs of size .

Is it true that ?

## Erdős–Faber–Lovász conjecture ★★★

Author(s): Erdos; Faber; Lovasz

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 .

Keywords: chromatic number

## Are there only finite Fermat Primes? ★★★

Author(s):

Conjecture   A Fermat prime is a Fermat number that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.

Keywords:

## 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 ,

## Erdős-Posa property for long directed cycles ★★

Author(s): Havet; Maia

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 .

Keywords:

## 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.

## 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: