did i do this correctly my work is below

Show that the following problem is in N P :

$1.$ Given a directed graph

G$,$ is there a directed circuit $($path that starts and ends on the same node $-$ i$.$e$.,$ a loop through

the graph$)$ that goes through each node exactly once.

Running time: O$($n$^2)$

$2.3$SAT $<=$ G

$3.$ Let $$ be a formula with $$ clauses. Generate an undirected graph $$:

For each clause in $,$ make a node for each literal. Make edge between every

pair of nodes, except:

$1.$ Nodes in the same clause

$2.$ Nodes that are negations of each other

$4.$ Yes. $($k $=$ num clauses$)$

$-3$k nodes

$-$ O$($K$^2)$ edges $($fewer

than complete graph$)$

$5.$ is satisfiable $$ has a $-$G$.$

$$ Suppose $$ is satisfiable. Then at least

one literal is true in each clause. For each

clause, select a node in $$ for one of the true

literals. This forms a K$-$G$,$ since K nodes

are selected and each is joined by an edge.

$6.$ Suppose $$ has a K$-$G Then there is

a non$-$contradictory node from the

$-$G in each clause.