(a) Let λ Z+. If we have λ different colors available, in how many ways can we...

(a) Let λ ˆˆ Z+. If we have λ different colors available, in how many ways can we color the vertices of the graph shown in Fig. 8.14(a) so that no adjacent vertices share the same color? This result in λ is called the chromatic polynomial of the graph, and the smallest value of λ for which the value of this polynomial is positive is called the chromatic number of the graph. What is the chromatic number of this graph? (We shall pursue this idea further in Chapter 11.)
(b) If there are six colors available, in how many ways can the rooms Ri, 1 ‰¤ i ‰¤ 5, shown in Fig. 8.14(b) be painted so that rooms with a common doorway, Dj, 1 ‰¤ j ‰¤ 5, are painted with different colors?

Transcribed Image Text:

R1 D3 Rs b) Figure 8.14