Coloring and Independent Sets
According to Brooks' theorem every cubic graph other than the complete graph K4 can be colored with at most three colors. Therefore, every cubic graph other than K4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices.
According to Vizing's theorem every cubic graph needs either three or four colors for an edge coloring. A 3-edge-coloring is known as a Tait coloring, and forms a partition of the edges of the graph into three perfect matchings. By König's line coloring theorem every bicubic graph has a Tait coloring.
The bridgeless cubic graphs that do not have a Tait coloring are known as snarks. They include the Petersen graph, Tietze's graph, the Blanuša snarks, the flower snark, the double-star snark, the Szekeres snark and the Watkins snark. There is an infinite number of distinct snarks.
Read more about this topic: Cubic Graph
Famous quotes containing the words independent and/or sets:
“Surrealism, n. Pure psychic automatism, by which it is intended to express, whether verbally or in writing, or in any other way, the real process of thought. Thought’s dictation, free from any control by the reason, independent of any aesthetic or moral preoccupation.”
—André Breton (1896–1966)
“In my dealing with my child, my Latin and Greek, my accomplishments and my money stead me nothing; but as much soul as I have avails. If I am wilful, he sets his will against mine, one for one, and leaves me, if I please, the degradation of beating him by my superiority of strength. But if I renounce my will, and act for the soul, setting that up as umpire between us two, out of his young eyes looks the same soul; he reveres and loves with me.”
—Ralph Waldo Emerson (1803–1882)