5. Suppose that G is a graph with chromatic polynomial $\pi_G$. Form a new graph H from
G by choosing an edge e of G and adding a new vertex z that is adjacent to the two
endvertices of e. Construct two more graphs G' and H' by deleting the edge e from G
and H, respectively. These graphs are illustrated in Figure 14: each of the graphs has
the same, unknown form within the ellipse.
(a) Explain why $\pi_H(k) = (k - 2)\pi_G(k)$.
(b) Give an example of a graph G and edge e so that $\pi_H(k) \neq (k - 2)\pi_G(k)$.
G
H
G'
[5]
[5]
H'
e
e
z
Figure 14: Graphs G, H, G' and H'.
z