Editing Chromatic polynomial (section)

===Chromatic roots===
A [[root of a function|root]] (or ''zero'') of a chromatic polynomial, called a “chromatic root”, is a value ''x'' where <math>P(G, x)=0</math>. Chromatic roots have been very well studied, in fact, Birkhoff’s original motivation for defining the chromatic polynomial was to show that for planar graphs, <math>P(G, x)>0</math> for ''x'' ≥ 4. This would have established the [[four color theorem]].

No graph can be 0-colored, so 0 is always a chromatic root. Only edgeless graphs can be 1-colored, so 1 is a chromatic root of every graph with at least one edge. On the other hand, except for these two points, no graph can have a chromatic root at a real number smaller than or equal to 32/27.<ref>{{harvtxt|Jackson|1993}}</ref> A result of Tutte connects the [[golden ratio]] <math>\varphi</math> with the study of chromatic roots, showing that chromatic roots exist very close to <math>\varphi^2</math>:
If <math>G_n</math> is a planar triangulation of a sphere then

:<math>P(G_n,\varphi^2) \leq \varphi^{5-n}.</math>

While the real line thus has large parts that contain no chromatic roots for any graph, every point in the [[complex plane]] is arbitrarily close to a chromatic root in the sense that there exists an infinite family of graphs whose chromatic roots are dense in the complex plane.<ref>{{harvtxt|Sokal|2004}}</ref>