Natural affordable bounded-degree property

Construct a natural bounded-degree monotone graph property of linear size that is affordable in a non-trivial way, meaning that it admits a successful strategy with budget equal to its size up to a subpolynomial factor and subquadratic exposure time, without arising from an artificial intersection of properties.

Background

The paper defines an affordable property as one admitting a successful strategy with budget s n{o(1)} and time at most n{2-Ω(1)}, where s is the minimum number of edges in the property. It gives an artificial affordable example formed by intersecting Hamiltonicity with containment of a K_4, then asks for a natural bounded-degree example that avoids this construction.

References

Does there exist a `natural' (in particular, which is not an intersection similar to the one described above) bounded-degree property $\mathcal{P}$ of linear size which is affordable in a non-trivial way?

— Graph factors and powers of Hamilton cycles in the budget-constrained random graph process  (2502.20870 - Díaz et al., 28 Feb 2025) in Problem, Section 7 “Final remarks and open problems”