# 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

## 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

## 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

## Matching cut and girth ★★

Author(s):

Question   For every does there exists a such that every graph with average degree smaller than and girth at least has a matching-cut?

Keywords: matching cut, matching, cut

## Strong 5-cycle double cover conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   Let be a circuit in a bridgeless cubic graph . Then there is a five cycle double cover of such that is a subgraph of one of these five cycles.

Keywords: cycle cover