Abstract
A perfect colouring Φ of a simple undirected connected graph G is an edge colouring such that each vertex is incident with exactly one edge of each colour. This paper concerns the problem of representing groups by graphs with perfect colourings. We define groups of graph automorphisms, which preserve the structure of the colouring, and characterize these groups up to isomorphism. Our considerations are based on the fact that every perfectly coloured graph is isomorphic to a Schreier coset graph on a group generated by involutions.
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