Decompositions of complete multipartite graphs (Q1043595)

Dobson and Marusic [\textit{E. Dobson} and \textit{D. Marušič}, ``An unusual decomposition of a complete 7-partite graph of order 28'', Discrete Math. 308, No.\,20, 4595--4598 (2008; Zbl 1153.05053)] asked if there exists an infinite family of graphs with imprimitive automorphism group each of which can be decomposed into at least two arc-transitive graphs, at least one of which having a primitive automorphism group, and at least one of which has an imprimitive automorphism group. They exhibited a specific graph (the complete 7-partite graph with 4 vertices in each part) having such a decomposition. This note answers their question by generalizing their construction to an infinite sequence of graphs, where the first graph in the sequence is the graph exhibited by Dobson and Marusic.