Question

4- Let G be a simple path of length 8. A valid coloring of the path is an assignment of colors to the vertices such that no edge is monochromatic (which means has both end points of the same color.) Compute the number of ways to color the path with six colors (red, green, blue, yellow, violet, black). There are no more restrictions.

Answer 1

Let  be a simple path with 8 vertices  such that the edges in  are precisely  .

Since there are 6 colors, we can choose any one of those 6 colors to assign $v_1$.

Once $v_1$ is assigned a color, $v_2$ can be assigned any of the remaining 5 colors so that $v_1v_2$ is not monochromatic.

Inductively, we see that for any  , once $v_i$ is assigned a color, $v_{i+1}$ can be assigned any of the remaining 5 colors so that $v_iv_{i+1}$ is not monochromatic.

Thus there are 6 choices of colors for $v_1$ and then successively, 5 choices of colors for each of $v_2,\ldots,v_8$. Thus the number of colorings is .