Unresolved orders in the (4,7)-spectrum

Determine whether 4-regular graphs of girth 7 exist at orders 69, 71, and 81 through 83.

Background

The (4,7)-cage has order 67, and the paper establishes an upper bound N(4,7)≤84. Several orders near the beginning of the spectrum and immediately below the eventual range remain unresolved, leaving the complete spectrum undetermined.

References

The existence of $(4,7)$-graphs of orders $69$, $71$, and $81$ to $83$ is still undecided.

— Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs  (2503.06466 - Eze et al., 9 Mar 2025) in Section 4.2, subsection “(4,7)-Spectrum”