Optimal strategies for fixed graphs

Determine asymptotically optimal successful strategies for constructing copies of every fixed graph in the budget-constrained random graph process.

Background

The paper notes that asymptotically optimal strategies are known for several classes of fixed graphs, including trees, cycles, diamonds, and certain collections of triangles. It explicitly identifies the remaining fixed-graph case as unresolved.

References

All other fixed graphs remain an open problem.

— Graph factors and powers of Hamilton cycles in the budget-constrained random graph process  (2502.20870 - Díaz et al., 28 Feb 2025) in Section 1, subsection “Previous results”