Existence of the order-37 (4,6)-graph

Determine whether a 4-regular graph of girth 6 and order 37 exists.

Background

The (4,6)-spectrum begins at order 26. The paper excludes several odd orders and establishes N(4,6)≤38, but order 37 remains the only unresolved order below the established eventual range discussed in this subsection.

References

The existence of a $(4,6)$-graph of order $37$ is currently undecided.

— Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs  (2503.06466 - Eze et al., 9 Mar 2025) in Section 4.1, subsection “(4,6)-Spectrum”