Connectivity parameters of Cartesian products of cycles

Determine m(C_m×C_n) for all positive integers m and n, where m(C_m×C_n) denotes the maximum size of a connectivity code in the Cartesian product of cycles C_m and C_n.

Background

The paper studies connectivity codes in regular graphs and completely determines the extremal function f(d), but it does not determine the corresponding maximum code sizes for all Cartesian products of cycles.

The unresolved problem concerns obtaining an exact formula for m(C_m×C_n) for arbitrary cycle lengths. The same sentence also identifies the complexity of computing or approximating m(H) for a given input graph as an unresolved issue, indicating an associated algorithmic problem beyond the exact evaluation for cycle products.

References

Determining $m(C_{m}\times C_{n})$ for all $m,n$ remains open , as does the complexity of computing or approximating $m(H)$ for a given input graph.

— Alon's Question on Connectivity Graph-Codes: $f(d)=2^d$ for Every $d\geq 4$  (2609.02953 - Tian, 2 Sep 2026) in Section 6, Concluding remarks, item (d)