How many solutions are there for the mapcoloring problem in Fig. $6\mathrm{.}1$ using three colors?

$($a$)$ the principal states and territories of Australia Coloring this map can be viewed as a constraint satisfaction problem $($CSP$).$ The goal is to assign color to each region so that no neighboring regions have the same color $($b$)$ the map$-$coloring problem represented as a constraint graph.