Symmetric bowtie decompositions of the complete graph (Q986704)

Summary: Given a bowtie decomposition of the complete graph \(K_v\) admitting an automorphism group \(G\) acting transitively on the vertices of the graph, we give necessary conditions involving the rank of the group and the cycle types of the permutations in \(G\). These conditions yield non-existence results for instance when \(G\) is the dihedral group of order \(2v\), with \(v\equiv 1,9 \pmod{12}\), or a group acting transitively on the vertices of \(K_9\) and \(K_{21}\). Furthermore, we have non-existence for \(K_{13}\) when the group \(G\) is different from the cyclic group of order 13 or for \(K_{25}\) when the group \(G\) is not an abelian group of order 25. Bowtie decompositions admitting an automorphism group whose action on vertices is sharply transitive, primitive or 1-rotational, respectively, are also studied. It is shown that if the action of \(G\) on the vertices of \(K_v\) is sharply transitive, then the existence of a \(G\)-invariant bowtie decomposition is excluded when \(v \equiv 9 \pmod{12}\) and is equivalent to the existence of a \(G\)-invariant Steiner triple system of order \(v\). We are always able to exclude existence if the action of \(G\) on the vertices of \(K_v\) is assumed to be 1-rotational. If, instead, \(G\) is assumed to act primitively then existence can be excluded when \(v\) is a prime power satisfying some additional arithmetic constraint.