# Arthur

## 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}

Keywords:

## 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

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

Author(s): Arthur; Hoffmann-Ostenhof

\begin{conjecture} Let $C$ be a circuit in a bridgeless cubic graph $G$. Then there is a five cycle double cover of $G$ such that $C$ is a subgraph of one of these five cycles. \end{conjecture}

Keywords: cycle cover