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
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