Universal non-bipartiteness threshold for even-girth graphs

Determine, for every integer k≥3 and every even integer g≥4, the smallest order n for which there exists a (k,g)-graph of order n that is not bipartite.

Background

All known cages and record graphs of even girth are bipartite, motivating the conjecture that every even-girth cage is bipartite. Independently of whether that conjecture is true, the authors formulate the problem of locating the first non-bipartite graph in each even-girth (k,g)-class.

References

This lead to the repeatedly stated conjecture that all even-girth cages must be bipartite. Whether the conjecture holds true or not, it still leads to the following open problem we find both interesting and related to $(k,g)$-spectra:

— Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs  (2503.06466 - Eze et al., 9 Mar 2025) in Concluding Remarks, Section 6