Write a function $`$encodeAndSolveThreeColoring$($n$,$ edge$\_$list$)`$ that given the number of vertices $n $\backslash$geq $1$$ and the list of edges as a list of pairs of vertices $`[($i$1,$ j$1),($i$2,$ j$2)...,($im$,$ jm$)]`$ returns a tuple $`($flag$,$ color$\_$assignment$)`$ consisting of boolean $`$flag$`$ and a list $`$color$\_$assignment$`,$ wherein

$-`$flag$`$ is $`$True$`$ if the graph is three colorable and $`$False$`$ if not.

$-`$color$\_$assignment$`$ is a list of n colors $`$r$`,`$g$`,$ or $`$b$`($standing for red, green or blue$)$ where the $i$^\{$th$\}$$ element of the list stands for the color assigned to vertex $i$$.$

Note that the $`$color$\_$assignment$`$ component of the return value is ignored if $`$flag$`$ is set to $`$False$`.$