# Recent Activity

## Is Skewes' number e^e^e^79 an integer? ★★

Author(s):

Conjecture

Skewes' number is not an integer.

Keywords:

## Minimal graphs with a prescribed number of spanning trees ★★

Author(s): Azarija; Skrekovski

Conjecture   Let be an integer and let denote the least integer such that there exists a simple graph on vertices having precisely spanning trees. Then ## Sticky Cantor sets ★★

Author(s):

Conjecture   Let be a Cantor set embedded in . Is there a self-homeomorphism of for every greater than so that moves every point by less than and does not intersect ? Such an embedded Cantor set for which no such exists (for some ) is called "sticky". For what dimensions do sticky Cantor sets exist?

Keywords: Cantor set

## Subgroup formed by elements of order dividing n ★★

Author(s): Frobenius

Conjecture

Suppose is a finite group, and is a positive integer dividing . Suppose that has exactly solutions to . Does it follow that these solutions form a subgroup of ?

Keywords: order, dividing

## Giuga's Conjecture on Primality ★★

Author(s): Giuseppe Giuga

Conjecture is a prime iff Keywords: primality

## Extremal problem on the number of tree endomorphism ★★

Author(s): Zhicong Lin

Conjecture   An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the vertices' trees, the star with vertices has the most endomorphisms, while the path with vertices has the least endomorphisms.

Keywords:

## Coloring the Odd Distance Graph ★★★

Author(s): Rosenfeld

The Odd Distance Graph, denoted , is the graph with vertex set and two points adjacent if the distance between them is an odd integer.

Question   Is ?

Keywords: coloring; geometric graph; odd distance

## Cores of Cayley graphs ★★★★★

Author(s): Samal

Conjecture   Let be an abelian group. Is the core of a Cayley graph (on some power of ) a Cayley graph (on some power of )?

Keywords: Cayley graph; core

## Triangle free strongly regular graphs ★★★

Author(s):

Problem   Is there an eighth triangle free strongly regular graph?

Keywords: strongly regular; triangle free

## Graph product of multifuncoids ★★

Author(s): Porton

Conjecture   Let is a family of multifuncoids such that each is of the form where is an index set for every and is a set for every . Let every for some multifuncoid of the form regarding the filtrator . Let is a graph-composition of (regarding some partition and external set ). Then there exist a multifuncoid of the form such that regarding the filtrator .

Keywords: graph-product; multifuncoid

## Atomicity of the poset of multifuncoids ★★

Author(s): Porton

Conjecture   The poset of multifuncoids of the form is for every sets and :
\item atomic; \item atomistic.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords: multifuncoid

## Atomicity of the poset of completary multifuncoids ★★

Author(s): Porton

Conjecture   The poset of completary multifuncoids of the form is for every sets and :
\item atomic; \item atomistic.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords: multifuncoid

## Cycle double cover conjecture ★★★★

Author(s): Seymour; Szekeres

Conjecture   For every graph with no bridge, there is a list of cycles so that every edge is contained in exactly two.

Keywords: cover; cycle

## Upgrading a completary multifuncoid ★★

Author(s): Porton

Let be a set, be the set of filters on ordered reverse to set-theoretic inclusion, be the set of principal filters on , let be an index set. Consider the filtrator .

Conjecture   If is a completary multifuncoid of the form , then is a completary multifuncoid of the form .

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords:

## 4-regular 4-chromatic graphs of high girth ★★

Author(s): Grunbaum

Problem   Do there exist 4-regular 4-chromatic graphs of arbitrarily high girth?

Keywords: coloring; girth

## Perfect cuboid ★★

Author(s):

Conjecture   Does a perfect cuboid exist?

Keywords:

## Forcing a $K_6$-minor ★★

Author(s): Barát ; Joret; Wood

Conjecture   Every graph with minimum degree at least 7 contains a -minor.
Conjecture   Every 7-connected graph contains a -minor.

Keywords: connectivity; graph minors

## Funcoidal products inside an inward reloid ★★

Author(s): Porton

Conjecture   (solved) If then for every funcoid and atomic f.o. and on the source and destination of correspondingly.

A stronger conjecture:

Conjecture   If then for every funcoid and , .

Keywords: inward reloid

## Odd cycles and low oddness ★★

Author(s):

Conjecture   If in a bridgeless cubic graph the cycles of any -factor are odd, then , where denotes the oddness of the graph , that is, the minimum number of odd cycles in a -factor of .

Keywords:

## Odd perfect numbers ★★★

Author(s): Ancient/folklore

Conjecture   There is no odd perfect number.

Keywords: perfect number