Other Properties
The pathwidth of any n-vertex cubic graph is at most n/6. However, the best known lower bound on the pathwidth of cubic graphs is smaller, 0.082n.
It follows from the handshaking lemma, proven by Leonhard Euler in 1736 as part of the first paper on graph theory, that every cubic graph has an even number of vertices.
Petersen's theorem states that every cubic bridgeless graph has a perfect matching. Lovász and Plummer conjectured that every cubic bridgeless graph has an exponential number of perfect matchings. The conjecture was recently proved, showing that every cubic bridgeless graph with n vertices has at least 2n/3656 perfect matchings.
Read more about this topic: Cubic Graph
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)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)