Determine the stabilization threshold and eventual gonality of even-regular Harary graphs

Determine the integer n_k and the stabilized gonality value gon(H_{2k,n_k}) for each positive integer k in the theorem asserting eventual stabilization of the gonality of the 2k-regular Harary graphs H_{2k,n}.

Background

The paper proves that for every positive integer k, the gonality of the 2k-regular Harary graphs H_{2k,n} eventually becomes constant as n increases. However, the result is non-effective: it establishes existence of a threshold n_k but does not identify either that threshold or the common eventual gonality value. The authors later make the result effective only for k=2, proving that the 4-regular Harary graph H_{4,n} has gonality 10 for n 16.

References

Note that this result is not effective, in that we do not know the value of n_k or of \gon(H_{2k},n_k).

— The gonality of circulant graphs  (2508.05761 - Cenek et al., 7 Aug 2025) in Section 1, immediately following the theorem on stabilization of H_{2k,n}