Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / R / 1 / d / Solution

Use the convention that $W_n$ is the wheel graph with rim $C_n$ and one additional hub; the assumption $n\ge 3$ gives the rim at least three vertices. The rim needs two colours for even $n$ and three for odd $n$. Since the hub is adjacent to every rim vertex, its colour must be a new one. These lower bounds are attained by properly colouring the rim and adding that extra colour. ThusIf a convention indexes a wheel by its total number of vertices instead, replace the rim size $n$ by $n-1$ in this formula.