Question:
Let $R$ be the relation $\{(1,1),(2,3),(2,2),(3,2),(3,3)\}$ on the set $S=\{1,2,3\}$. Is $R$ an equivalence relation? If $R$ is, describe the partition $\mathscr{P}$ determined by $R$ by listing the pieces in $\mathscr{P}$.
Attempt:
$R$ is reflexive since for all $s\in S$ we have that $sRs$. $R$ is symmetric since each pair $(x,y)\in R$ we have that $(y,x)\in R$ also, namely for all $s,s'\in S$ we have that $sRs'\implies s'Rs$. Lastly, and this is what I'm not sure of, $R$ is transitive because the subgraphs $G_1$ and $G_{2,3}$ of the graph $G$ (see below) are Eulerian:
Except the proof of the transitivity, all you proof are correct. But Eulerness of the relation graph is not a sufficient conditon of transitivity of the relation, as shows a relation $\{(1,2), (2,3),(3,1)\}$ on the same set $S$.