Let $n$ be a positive integer, and let $p(n)$ be the number of partitions of $n$. For two partitions $p_1, p_2$ of the same integer $n$, we say that $p_2$ is a refinement of $p_1$ if the parts of $p_1$ can be subdivided to produce the parts of $p_2$. For example, $(2, 2, 1)$ is a refinement of $(4, 1)$ but $(2, 2, 1)$ is not a refinement of $(3, 1, 1)$. For each positive integer $n$, define a graph $G_n$ whose vertices are the partitions of n, so $|V (G_n)| = p(n)$, and whose edge set is $$E(G_n) = \{{u, v} : u \ne v, \,\,\text{and}\,\, u \text{ is a refinement of } v \text{ or } v \text{ is a refinement of } u\}$$ Prove that $G_n$ is Eulerian if and only if $n = 1$ or $n = 3$.
I’ve tried drawing out the graphs $n=4$ and $n=5$ but haven’t made any connections. I’m not sure how one should approach a question like this.
Any help appreciated, thank you :).