Automorphism groups of graphs with 1-factorizations Original Research Article

A 1-factorization 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. The automorphisms which preserve the colours of all edges constitute a group Ac (G, ϕ). We prove every finitely generated group H to be isomorphic to the full group Ac(G, ϕ) for a regular graph G of degree 3 with a 1-factorization ϕ. Moreover we show that for every finitely generated group H there is a regular graph G of degree 5 such that the group H and all of its subgroups can be represented (up to isomorphism) by a group of colour preserving automorphisms related to some 1-factorization ϕ of G.