Let $G$ be a group, and fix a symmetric generating set $S$. Let $X$ be the corresponding Cayley graph.

Let $R$ be a set of words in $S$, each corresponding to the identity of $G$, such that the set of closed walks (some authors would write `closed paths') in $X$ induced by the elements of $R$ generates $H_1(X)$, i.e. the first simplicial homology group over $\mathbb Z$. Is it true that $<S \mid R>$ is a presentation of $G$?