Existence of the order-34 (5,5)-graph

Determine whether a 5-regular graph of girth 5 and order 34 exists.

Background

The (5,5)-cage has order 30, and the paper determines many subsequent members of the spectrum, with N(5,5)≤36. The only potential order not yet decided in the reported range is 34; resolving it would complete the remaining gap in this spectrum.

References

The only potential member of the $(5,5)$-spectrum not yet decided is $34$.

— Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs  (2503.06466 - Eze et al., 9 Mar 2025) in Section 5.1, subsection “(5,5)-Spectrum”