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

Author(s): Arthur; Hoffmann-Ostenhof

\begin{conjecture} Let $G$ be a $2$-connected cubic graph and let $S$ be a $2$-regular subgraph such that $G-E(S)$ is connected. Then $G$ has a cycle double cover which contains $S$ (i.e all cycles of $S$). \end{conjecture}

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

Author(s): Sands; Sauer; Woodrow

\begin{conjecture} For every $k$, there exists an integer $f(k)$ such that if $D$ is a digraph whose arcs are colored with $k$ colors, then $D$ has a $S$ set which is the union of $f(k)$ stables sets so that every vertex has a monochromatic path to some vertex in $S$. \end{conjecture}

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## 3-Decomposition Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

\begin{conjecture} (3-Decomposition Conjecture) Every connected cubic graph $G$ has a decomposition into a spanning tree, a family of cycles and a matching. \end{conjecture}

Keywords: cubic graph

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

Author(s): Porton

Warning: This formulation is vague (not exact).

\begin{question} Characterize the set $\{f\in\mathsf{FCD} \mid (\mathsf{RLD})_{\mathrm{in}} f=(\mathsf{RLD})_{\mathrm{out}} f\}$. In other words, simplify this formula. \end{question}

The problem seems rather difficult.

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## A diagram about funcoids and reloids ★★

Author(s): Porton

Define for posets with order $\sqsubseteq$:

1. $\Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$;
2. $\Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigcap \{ x \in \mathfrak{B} \mid f x \sqsupseteq b \}$.

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

Then we have the following diagram:

\Image{diagram2.png}

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

\begin{conjecture} "Other" is $\lambda f\in\mathsf{FCD}: \top$. \end{conjecture}

\begin{question} What repeated applying of $\Phi_{\ast}$ and $\Phi^{\ast}$ to "other" leads to? Particularly, does repeated applying $\Phi_{\ast}$ and/or $\Phi^{\ast}$ to the node "other" lead to finite or infinite sets? \end{question}

Keywords: Galois connections

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

Author(s): Porton

\begin{conjecture} For every composable funcoids $f$ and $g$ $$(\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq(\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{RLD})_{\mathrm{out}}f.$$ \end{conjecture}

Keywords: outward reloid

## A funcoid related to directed topological spaces ★★

Author(s): Porton

\begin{conjecture} Let $R$ be the complete funcoid corresponding to the usual topology on extended real line $[-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\}$. Let $\geq$ be the order on this set. Then $R\sqcap^{\mathsf{FCD}}\mathord{\geq}$ is a complete funcoid. \end{conjecture}

\begin{proposition} It is easy to prove that $\langle R\sqcap^{\mathsf{FCD}}\mathord{\geq}\rangle \{x\}$ is the infinitely small right neighborhood filter of point $x\in[-\infty,+\infty]$. \end{proposition}

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

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