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-edge-coloring or a dominating cycle), which are known to be equivalent, are equivalent to 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 Ryjáček and Schelp in 2003 as a common generalization and strengthening of the reduction techniques by Catlin and Veldman and of the closure concept introduced by Ryjáček in 1997.