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:
“Men will say that in supporting their wives, in furnishing them with houses and food and clothes, they are giving the women as much money as they could ever hope to earn by any other profession. I grant it; but between the independent wage-earner and the one who is given his keep for his services is the difference between the free-born and the chattel.”
—Elizabeth M. Gilmer (1861–1951)
“bars of that strange speech
In which each sound sets out to seek each other,
Murders its own father, marries its own mother,
And ends as one grand transcendental vowel.”
—Randall Jarrell (1914–1965)