(Solution) - For n 3 recall that the wheel graph Wn -(2025 Original AI-Free Solution)

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For n ? 3, recall that the wheel graph, Wn, is obtained from a cycle of length n by placing a new vertex within the cycle and adding edges (spokes) from this new vertex to each vertex of the cycle.
(a) What relationship is there between x (Cn) and x (Wn)?
(b) Use part (e) of Exercise 15 to show that
P(Wn, ?) = ?(? - 2)n + (-1 )n?(? - 2).
(c) (i) If we have k different colors available, in how many ways can we paint the walls and ceiling of a pentagonal room if adjacent walls, and any wall and the ceiling, are to be painted with different colors?
(ii) What is the smallest value of k for which such a coloring is possible?