Definition
The chromatic polynomial of a graph counts the number of its proper vertex colorings. It is commonly denoted, or, and sometimes in the form, where it is understood that for fixed the function is a polynomial in, the number of colors.
For example, the path graph on 3 vertices cannot be colored at all with 0 or 1 colors. With 2 colors, it can be colored in 2 ways. With 3 colors, it can be colored in 12 ways.
| Available colors | 0 | 1 | 2 | 3 |
| Number of colorings | 0 | 0 | 2 | 12 |
The chromatic polynomial is defined as the unique interpolating polynomial of degree through the points for, where is the number of vertices in . For the example graph, and indeed .
The chromatic polynomial includes at least as much information about the colorability of as does the chromatic number. Indeed, the chromatic number is the smallest positive integer that is not a root of the chromatic polynomial,
Read more about this topic: Chromatic Polynomial
Famous quotes containing the word definition:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)