Graph Coloring · Superchallenge problem

The icosahedron graph has $12$ vertices (each of degree $5$, $30$ edges total): $6$ vertices $1,\ldots,6$ forming a "cap" around vertex... concretely, edges $12,13,14,15,16,23,34,45,56,62$ (hub $1$ joined to a $5$-cycle $2,3,4,5,6$), a parallel ring $7,\ldots,11$ with edges $78,89,9\text{-}10,10\text{-}11,11\text{-}7$, cross edges $27,38,49,5\text{-}10,6\text{-}11$ and $28,39,4\text{-}10,5\text{-}11,67$, and a second hub $12$ joined to all of $7,\ldots,11$. What is its chromatic number?