$($a$)$ Construct a small $3-$colorable graph with $7$ vertices with at least one $4-$degree vertex, at most

two $2-$degree vertices, and no $1-$degree vertices.

$($b$)$ Draw the building blocks for one vertex of degree $4$ or more $($as in Paper $2)$ and one example of

a building block of an edge and briefly explain how these building blocks are constructed.

$($c$)$ Write down the algorithm for the $3-$colorability problem and give an example of one structure that

will be discarded at each step.

$($d$)$ Can you draw a covering graph of your graph? In general, should covering graphs be discarded

for this algorithm? Why or why not?