Chromatic polynomial

Definition

Let $G$ be a finite graph. For each natural number $k$ , let $P(G, k)$ be the number of colourings of $G$ with the set of colours $\{1, \dots, k\}$ . There is a unique polynomial $P(G, x)$ whose value at each natural number $k$ is $P(G, k)$ . It is the chromatic polynomial of $G$ .

Remarks

For the path graph $P_3$ , $P(P_3, x) = x(x-1)^2$ . For the complete graph, $P(K_n, x) = x(x-1)\cdots(x-n+1)$ . For an edge $e$ , the deletion–contraction rule says $P(G, x) = P(G - e, x) - P(G / e, x)$ . Here $G / e$ merges the two ends of $e$ . The chromatic number is the smallest positive integer $k$ with $P(G, k) > 0$ .

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Graph colouring definition
Chromatic polynomial definition

Tag	`038YGY0HSC`
URI	`m2://038YGY0HSC`
Kind	definition
Writing	W1 stated
MSC	05C31, 05C15
Defines	Chromatic polynomial
Uses	Graph colouring

Chromatic polynomial

Definition

Remarks

Reading path

Further reading

What this page needs for W2

What links here