Symmetric bowtie decompositions of the complete graph

Given a bowtie decomposition of the complete graph Kvadmitting an automorphism group G acting transitively on thevertices of the graph, we give necessary conditions involvingthe rank of the group and the cycle types of the permutationsin G. These conditions yield non--existence results forinstance when G is the dihedral group of order 2v, with$v\equiv 1, 9\pmod{12}$, or a group acting transitively on thevertices of K9 and K_{21}. Furthermore, we havenon--existence for K_{13} when the group G is differentfrom the cyclic group of order 13 or for K_{25} when thegroup G is not an abelian group of order 25. Bowtiedecompositions admitting an automorphism group whose action onvertices is sharply transitive, primitive or 1--rotational,respectively, are also studied. It is shown that if the actionof G on the vertices of K_v is sharply transitive, then theexistence of a G--invariant bowtie decomposition is excludedwhen $v\equiv 9\pmod{12}$ and is equivalent to the existence ofa G--invariant Steiner triple system of order v. We arealways able to exclude existence if the action of G on thevertices of K_v is assumed to be 1--rotational. If,instead, G is assumed to act primitively then existence canbe excluded when v is a prime power satisfying someadditional arithmetic constraint.