Unresolved orders in the cubic (3,10)-spectrum

Determine whether cubic graphs of girth 10 exist at each of the orders 74, 76, and 78.

Background

The paper determines the (3,10)-cage order as 70 and establishes graphs at many larger orders, with N(3,10)≤80. The existence status of orders 74, 76, and 78 remains unresolved; excluding one would create another spectral hole, while excluding two would produce two holes.

References

The existence or non-existence of $(3,10)$-graphs of orders $ \{74, 76, 78 \} $ is yet to be determined.

— Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs  (2503.06466 - Eze et al., 9 Mar 2025) in Section 3.1, subsection “(3,10)-Spectrum”