Graphs with unbounded \(\tilde{\nu}\)-rank

Construct a family of graphs whose \(\tilde{\nu}\)-rank is unbounded.

Background

The paper constructs graph classes for which the proposed hierarchy has rank one, while related hierarchies require many levels. It observes that the rank cannot be universally bounded by a constant unless SDP feasibility is NP-hard, and reports that the examples obtained so far have rank at most two. The existence of graphs with arbitrarily large rank is left unresolved.

References

So far, we have only produced graphs with $$ at most 2. It remains an open problem to find a family of graphs with unbounded $$.

— Low degree sum-of-squares bounds for the stability number: a copositive approach  (2509.04949 - Vargas et al., 5 Sep 2025) in Section 10, Conclusion and Discussion