Graph Coloring · Hard problem

Two pentagons meet at exactly one shared vertex: the first pentagon has vertices $0,1,2,3,4$ with edges $0\text{-}1\text{-}2\text{-}3\text{-}4\text{-}0$, and the second has vertices $0,5,6,7,8$ with edges $0\text{-}5\text{-}6\text{-}7\text{-}8\text{-}0$ ($9$ vertices, $10$ edges total). With $3$ colors available, in how many different ways can all vertices be assigned colors so that every edge joins two different colors?