# Recent Activity

## Cycle Double Covers Containing Predefined 2-Regular Subgraphs ★★★

Author(s): Arthur; Hoffmann-Ostenhof

**Conjecture**Let be a -connected cubic graph and let be a -regular subgraph such that is connected. Then has a cycle double cover which contains (i.e all cycles of ).

Keywords:

## Monochromatic reachability in arc-colored digraphs ★★★

Author(s): Sands; Sauer; Woodrow

**Conjecture**For every , there exists an integer such that if is a digraph whose arcs are colored with colors, then has a set which is the union of stables sets so that every vertex has a monochromatic path to some vertex in .

Keywords:

## 3-Decomposition Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

**Conjecture**(3-Decomposition Conjecture) Every connected cubic graph has a decomposition into a spanning tree, a family of cycles and a matching.

Keywords: cubic graph

## Which outer reloids are equal to inner ones ★★

Author(s): Porton

Warning: This formulation is vague (not exact).

**Question**Characterize the set . In other words, simplify this formula.

The problem seems rather difficult.

Keywords:

## A diagram about funcoids and reloids ★★

Author(s): Porton

Define for posets with order :

- ;
- .

Note that the above is a generalization of monotone Galois connections (with and replaced with suprema and infima).

Then we have the following diagram:

What is at the node "other" in the diagram is unknown.

**Conjecture**"Other" is .

**Question**What repeated applying of and to "other" leads to? Particularly, does repeated applying and/or to the node "other" lead to finite or infinite sets?

Keywords: Galois connections

## Outward reloid of composition vs composition of outward reloids ★★

Author(s): Porton

**Conjecture**For every composable funcoids and

Keywords: outward reloid

## Sum of prime and semiprime conjecture ★★

Author(s): Geoffrey Marnell

**Conjecture**Every even number greater than can be represented as the sum of an odd prime number and an odd semiprime .

## A funcoid related to directed topological spaces ★★

Author(s): Porton

**Conjecture**Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.

**Proposition**It is easy to prove that is the infinitely small right neighborhood filter of point .

If proved true, the conjecture then can be generalized to a wider class of posets.

Keywords:

## Infinite distributivity of meet over join for a principal funcoid ★★

Author(s): Porton

**Conjecture**for principal funcoid and a set of funcoids of appropriate sources and destinations.

Keywords: distributivity; principal funcoid

## Weak saturation of the cube in the clique ★

**Problem**

Determine .

Keywords: bootstrap percolation; hypercube; Weak saturation

## Convex Equipartitions with Extreme Perimeter ★★

Author(s): Nandakumar

To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.

Remark: It appears maximizing the total perimeter is the easier problem.

Keywords: convex equipartition

## Turán Problem for $10$-Cycles in the Hypercube ★★

Author(s): Erdos

**Problem**Bound the extremal number of in the hypercube.

Keywords: cycles; extremal combinatorics; hypercube

## Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★

**Problem**Determine the smallest percolating set for the -neighbour bootstrap process in the hypercube.

Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation

## Saturation in the Hypercube ★★

Author(s): Morrison; Noel; Scott

**Question**What is the saturation number of cycles of length in the -dimensional hypercube?

Keywords: cycles; hypercube; minimum saturation; saturation

## Cycles in Graphs of Large Chromatic Number ★★

Author(s): Brewster; McGuinness; Moore; Noel

**Conjecture**If , then contains at least cycles of length .

Keywords: chromatic number; cycles

## The Double Cap Conjecture ★★

Author(s): Kalai

**Conjecture**The largest measure of a Lebesgue measurable subset of the unit sphere of containing no pair of orthogonal vectors is attained by two open caps of geodesic radius around the north and south poles.

Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere

## Circular flow numbers of $r$-graphs ★★

Author(s): Steffen

A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and .

A -regular graph is a -graph if for every with odd.

**Conjecture**Let be an integer. If is a -graph, then .

Keywords: flow conjectures; nowhere-zero flows

## Circular flow number of regular class 1 graphs ★★

Author(s): Steffen

A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and . The circular flow number of is inf has a nowhere-zero -flow , and it is denoted by .

A graph with maximum vertex degree is a class 1 graph if its edge chromatic number is .

**Conjecture**Let be an integer and a -regular graph. If is a class 1 graph, then .

## Chromatic number of associahedron ★★

Author(s): Fabila-Monroy; Flores-Penaloza; Huemer; Hurtado; Urrutia; Wood

**Conjecture**Associahedra have unbounded chromatic number.

## Are there infinite number of Mersenne Primes? ★★★★

Author(s):

**Conjecture**A Mersenne prime is a Mersenne number that is prime.

Are there infinite number of Mersenne Primes?

Keywords: Mersenne number; Mersenne prime