## 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 :

1. ;
2. .

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

## 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