Switching $(m, n)$-mixed graphs with respect to Abelian groups

We extend results of Brewster and Graves for switching $m$-edge coloured graphs with respect to a cyclic group to switching $(m, n)$-mixed graphs with respect to an Abelian group. In particular, we establish the existence of a $(m, n)$-mixed graph $P_\Gamma(H)$ with the property that a $(m, n)$-mixed graph $G$ is switch equivalent to $H$ if and only if it is a special subgraph of $P_\Gamma(H)$, and the property that that $G$ can be switched to have a homomorphism to $H$ if and only if it has a homomorphism (without switching) to $P_\Gamma(H)$. We consider the question of deciding whether a $(m, n)$-mixed graph can be switched so that it has a homomorphism to a proper subgraph, i.e. whether it can be switched so that it isn't a core. We show that this question is NP-hard for arbitrary groups and NP-complete for Abelian groups. Finally, we consider the complexity of the switchable $k$-colouring problem for $(m, n)$-mixed graphs and prove a dichotomy theorem in the cases where $m \geq 1$.