Questions, Problems, & Conjectures about the Chromatic Polynomial

From the 2005 ReSEARCH Group

1. Conjecture: For every graph that contains any amount of line segments and vertices in a properly colored graph in an equation there will always be one vertex that is raised by ‘x’ and all others variations of color will be represented by ‘x-1’, and depending on the amount of vertices’x-1’ will be raised to the preceding power. (R.F.)
Example: x(x-1)^ 3

G4

There is a 3 in the power. There are three line segments/three dots after the first point.

2. Conjectures:

G3
For triangles, the formula would be (x^2-x)(x-2).
For points without lines, (x^2-x)(nx)
For all quadrilaterals, (x^2-x) (x-2) (x-3)+3x+x is the number of proper colorings that use exactly x colors.
For polygons, (x^2-x)(x-2)[x-(n-1)], where n is the number of sides in the polygon. (R.B)

3. Question:

Could it (see conjectures below) work for any number? How can I prove it for any number? (C.T.),

4. Conjectures:
x(x-1) gives the total possible colorings for the graph G1, where x=# of colors.

G1

(A.F., C.T)

x(x-1)^ 2 or [ (x^2 –x)x]-2x gives the total possible colorings for the graph G2:

G2

(A.F, C.T.)