  We introduce the vertex-arboricity of group-labelled graphs. For an abelian
group $\Gamma$, a $\Gamma$-labelled graph is a graph whose edges are labelled
by elements of $\Gamma$. For an abelian group $\Gamma$ and $A\subseteq \Gamma$,
the $(\Gamma, A)$-vertex-arboricity of a $\Gamma$-labelled graph is the minimum
integer $k$ such that its vertex set can be partitioned into $k$ parts where
each part induces a subgraph having no cycle of value in $A$.
  We prove that for every positive integer $\omega$, there is a function
$f_{\omega}:\mathbb{N}\times\mathbb{N}\to \mathbb{R}$ such that if
$|\Gamma\setminus A|\le \omega$, then every $\Gamma$-labelled graph with
$(\Gamma, A)$-vertex-arboricity at least $f_{\omega}(t,d)$ contains a
subdivision of $K_t$ where all branching paths are of value in $A$ and of
length at least $d$. This extends a well-known result that every graph of
sufficiently large chromatic number contains a subdivision of $K_t$, in various
directions.
