Unbalanced complete multipartite graphs

Determine the minimum abelian Cayley-graph host order for unbalanced complete multipartite graphs.

Background

The paper gives a construction showing that an unbalanced complete multipartite graph K_{a_1,...,a_k} has host order at most k max_i a_i, but explicitly notes that this bound is not generally tight. For K_{1,1,5}, the local lower bound is 10 while the construction gives 15, and the paper explains why an obvious refinement does not achieve order 10. The exact value for unbalanced complete multipartite graphs is therefore left unresolved.

References

Determining \eta for unbalanced complete multipartite graphs is open; the representation-number analogue was settled in .

— Induced Embeddings of Graphs into Abelian Cayley Graphs  (2609.01486 - Souop et al., 1 Sep 2026) in Remark 3.4, Section 3 (Complete multipartite graphs: an upper bound only)