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Let $G$ be a graph whose edges are each assigned one of the $m$-colours $1,2, \ldots, m$, and let $\Gamma$ be a subgroup of $S_m$. The operation ofswitching at a vertex $x$ with respect $\pi \in \Gamma$ permutes the colours ofthe edges incident with $x$ according to $\pi$. There is a well-developedtheory of switching when $\Gamma$ is Abelian. Much less is known fornon-Abelian groups. In this paper we consider switching with respect tonon-Abelian groups including symmetric, alternating and dihedral groups. Wefirst consider the question of whether there is a sequence of switches usingelements of $\Gamma$ that transforms an $m$-edge-coloured graph $G$ to an$m$-edge coloured graph $H$. Necessary and sufficient conditions for theexistence of such a sequence are given for each of the groups being considered.We then consider the question of whether an $m$-edge coloured graph can beswitched using elements of $\Gamma$ so that the transformed $m$-edge colouredgraph has a vertex $k$-colouring, or a homomorphism to a fixed $m$-edgecoloured graph $H$. For the groups just mentioned we establish dichotomytheorems for the complexity of these decision problems. These are the firstdichotomy theorems to be established for colouring or homomorphism problems andswitching with respect to any group other than $S_2$.

Switching $m$-edge-coloured graphs using non-Abelian groups