Linear-size free sets in bounded-degree planar graphs (Δ ≤ 6)

Determine whether every n-vertex planar graph with maximum degree Δ ∈ {3,4,5,6} contains a free set—i.e., a vertex subset S that can be mapped to any set P of |S| points while admitting a straight-line crossing-free drawing—of size Ω(n).

Background

Linear-sized free sets are known for several subclasses (trees, outerplanar graphs, Halin graphs, squaregraphs) and for certain degree-3 triconnected planar graphs. However, constructions via shortness exponents show that linear free sets cannot hold universally for maximum degree 7.

The status for Δ ∈ {3,4,5,6} remains unresolved; establishing linear lower bounds would bridge combinatorial constraints (bounded degree) with geometric embeddability (free sets).