Graph Coloring · Hard problem

A wheel graph $W_5$ has a central hub vertex $0$ directly joined to all five vertices $1,2,3,4,5$ of a pentagon rim, where the rim vertices themselves form the cycle $1\text{-}2\text{-}3\text{-}4\text{-}5\text{-}1$ ($6$ vertices, $10$ edges total). Brooks' theorem applies to this graph since it is connected, not complete, and not an odd cycle. By how much does Brooks' bound $\Delta(G)$ exceed the TRUE chromatic number of this wheel?