Abstract
Summary: A perfect colouring $\varphi$ 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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