  Let $S_g$ denote a closed, orientable surface of genus $g \geq 2$ and
$\mathcal{C}(S_g)$ be the associated curve graph. Let $d$ be the path metric on
$\mathcal{C}(S_g)$ and $a_0$ and $a_4$ be a pair of curves on $S_g$ with
$d(a_0, a_4) = 4$. In this article, we fix the vertex $a_0$ and apply the Dehn
twist about $a_4$, $T_{a_4}$, to it in an attempt to create pairs of curves at
a distance $5$ apart. We give a necessary and sufficient topological condition
for $d(a_0, T_{a_4}(a_0))$ to be $4$. We then characterise the pairs of $a_0$
and $a_4$ for which $5 \leq d(a_0, T_{a_4}(a_0)) \leq 6$. Lastly, we give an
example of a pair of curves on $S_2$ which represent vertices at a distance $5$
in $\mathcal{C}(S_2)$ with intersection number $144$.
