Cartesian-product sharpness

Determine whether η(G_1□G_2) can be substantially smaller than η(G_1)η(G_2), and determine whether equality in the Cartesian-product bound characterizes a natural class of graphs.

Background

The paper proves η(G_1□G_2)≤η(G_1)η(G_2). For path grids this yields asymptotically linear host orders, but the bound is not always exact: η(P_3□P_3)=15 while the product bound gives 16. The extent of possible improvement and any structural characterization of equality remain unresolved.

References

Is \eta(G_1\square G_2) ever much smaller than the bound of Theorem~\ref{thm:product}, and does equality characterise any natural class?

— Induced Embeddings of Graphs into Abelian Cayley Graphs  (2609.01486 - Souop et al., 1 Sep 2026) in Problem 2 (Cartesian products), Section 10 (Open problems)

Find a lower bound on \eta sensitive to more than one neighbourhood at a time.

— Induced Embeddings of Graphs into Abelian Cayley Graphs  (2609.01486 - Souop et al., 1 Sep 2026) in Problem 7 (Sharpening the floor), Section 10 (Open problems)