Contractible subgraphs, Thomassen's conjecture and the dominating cycle conjecture for snarks

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is hamiltonian), by Thomassen (every 4-connected line graph is hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edgecoloring or a dominating cycle), which are known to be equivalent, are equivalent with the statement that every snark (i.e. a cyclically 4-edge-connected cubic graph of girth at least five that is not 3-edge-colorable) has a dominating cycle. We use a refinement of the contractibility technique which was introduced by Ryj

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Hajo Broersma, Gasper Fijavz, Tomás Kaiser, Roman Kuzel, Zdenek Ryjácek, Petr Vrána
• Keywords: 4-connected Claw-free Graph, 4-edge-connected Cubic Graph, DM 2008, Dominating Cycle