Properties
All snarks are non-Hamiltonian, and many known snarks are hypohamiltonian: the removal of any single vertex leaves a Hamiltonian subgraph. A hypohamiltonian snark must be bicritical: the removal of any two vertices leaves a 3-edge-colorable subgraph. While snarks may be elusive, they are not rare: Z.Skupień proved that there are exponentially many hypohamiltonian snarks.
The cycle double cover conjecture posits that in every bridgeless graph one can find a collection of cycles covering each edge twice, or equivalently that the graph can be embedded onto a surface in such a way that all faces of the embedding are simple cycles. Snarks form the difficult case for this conjecture: if it is true for snarks, it is true for all graphs. In this connection, Branko Grünbaum conjectured that it was not possible to embed any snark onto a surface in such a way that all faces are simple cycles and such that every two faces either are disjoint or share only a single edge; however, a counterexample to Grünbaum's conjecture was found by Martin Kochol.
Read more about this topic: Snark (graph Theory)
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
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—Ralph Waldo Emerson (1803–1882)