Arthur


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

Author(s): Arthur; Hoffmann-Ostenhof

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 $).

Keywords:

3-Decomposition Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   (3-Decomposition Conjecture) Every connected cubic graph $ G $ 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 $ 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.

Keywords: cycle cover

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