Let $G$ be a group and let $\Gamma(G)$ be the commuting graph of $G$, with vertex set $G$ and $x\sim y$ if and only if $xy=yx$. Let $h(n)$ be minimal such that if the largest independent set of $\Gamma(G)$ has at most $n$ elements then $\Gamma(G)$ can be covered by $h(n)$ many complete subgraphs. Estimate $h(n)$ as well as possible.

#117: [Er90]

If $G$ is an abelian group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)

#274: [ErGr80]

group theory, covering systems