Question

2. Use the formula Pa(k) = PG-e(k) - PG/e(k) to find the chromatic polynomial of the following graph (take e as shown).

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Answer 1

Sol: Here, first we draw the two graphs G-e and G/e. Then we find the chromatic polynomial of the two graphs. Finally, we apply the given formula to find the chromatic polynomial of the given graph G.
Solh k -) k- G-e - Gle Here, k is the chromatic number, The number of possible colours for each of the vertices of the graphs G-l and Gle are shown in the figure. in the chromatic polynomial of Ge is, B. (k) = K.K. (k-1), (k-1).(k-1). =k (k-1)3 the chromatic polynomial of Gle is, Powe(k) k. (k-1). (k-1), (k-1) = K (k-1)}

in the chromatic polynomial of the given graph G من PG (k) = Pare (k) - Pale(K) =k? (k-1)} – k(k-1)} = (k-1)} (ki-k) = (k-1). K. (K-1) =k. (k-14

Verifications k e k -1 k- k -) Pa(k) - K. (k-1). (k-1), (k-1)(k-1) -k(k-1)"