Let $G$ be a graph in which each edge is assigned one of the colours $1, 2, \ldots, m$, and let $\Gamma$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ of $G$ with respect to an element $\pi$ of $\Gamma$ permutes the colours of the edges incident with $x$ according to $\pi$. We investigate the complexity of whether there exists a sequence of switches that transforms a given $m$-edge coloured graph $G$ so that it has a colour-preserving homomorphism to a fixed $m$-edge coloured graph $H$ and give a dichotomy theorem in the case that $\Gamma$ acts transitively.

翻译：设$G$是一个每条边被赋予颜色$1, 2, \ldots, m$之一的图，并设$\Gamma$为$S_m$的一个子群。在顶点$x$处根据$\Gamma$中元素$\pi$进行切换的操作，会按$\pi$置换与$x$关联的边的颜色。我们研究是否存在一系列切换操作，使得给定的$m$边着色图$G$变换后，对固定的$m$边着色图$H$具有保色同态的问题的复杂性，并在$\Gamma$传递作用的情况下给出了一个二分性定理。