Abundant regular Mycielskians

Determine whether the constructed abundant \(d\)-regular graph is always the unique abundant \(d\)-regular graph, and determine how large \(\gp(\mathcal{M}(G))-n\) can be for \(d\)-regular graphs.

Background

For regular graphs, the survey discusses bounds on the general position number of the Mycielskian and a construction of an abundant dd-regular graph of order $3d-1$. Uniqueness is known only for d=2,3d=2,3, leaving both uniqueness in general and the maximum excess over the order unresolved.

References

It is an open question whether this construction is always the unique abundant $d$-regular graph and, if not, how large the difference $\gp (\mathcal{M}(G))-n$ can be for $d$-regular graphs.

— The General Position Problem: A Survey  (2501.19385 - V. et al., 31 Jan 2025) in Section 3, subsection “Mycielskian and double graphs”