cluster in which it can participate is the one obtained by combining the critical row with the six (type \[type:AAA\]) rows of the form $\langle 0, u, u\rangle$.\
[[*Proof:* ]{}]{}Clearly, the critical row must pay a cost of $1$, since it has a special symbol. If it pays no extra cost, then the rows it is combined with should have the symbol ’$u$’ in their second and third columns. There are exactly six such rows available and these are the (type \[type:AAA\]) rows of the form $\langle 0, u, u\rangle$. [$\Box$]{}

We say that a vertex is [*perfect*]{}, if all the 20 rows corresponding to it pay an extra cost of 0. A vertex is said to be [*imperfect*]{}, if at least one of its 20 rows pay an extra cost of at least 1.

[Claim 3: ]{}Consider an edge $(x, y)$. If both $x$ and $y$ are perfect, then at least one of the two edge rows corresponding to the edge pays a cost of at least 2.\
[[*Proof:* ]{}]{}Consider the edge row $\langle 0, x, y\rangle$, corresponding to the given edge. Let $S$ be the cluster to which this row belongs. Since $k=7$, we have $|S|\geq 7$. Recall 