# Recent Activity

## 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 ★★★★

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

## 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?

## 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

## Characterizing (aleph_0,aleph_1)-graphs ★★★

Call a graph an -*graph* if it has a bipartition so that every vertex in has degree and every vertex in has degree .

**Problem**Characterize the -graphs.

Keywords: binary tree; infinite graph; normal spanning tree; set theory

## The Berge-Fulkerson conjecture ★★★★

**Conjecture**If is a bridgeless cubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .

Keywords: cubic; perfect matching