# Arthur

## 3-Edge-Coloring Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

**Conjecture**Suppose with is a connected cubic graph admitting a -edge coloring. Then there is an edge such that the cubic graph homeomorphic to has a -edge coloring.

Keywords: 3-edge coloring; 4-flow; removable edge

## 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:

## 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

## 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