The forest-path partition conjecture for planar graphs without 4-cycles and 6-cycles

Let a planar graph be a graph that can be embedded in the plane, and let a graph be forest if it is acyclic. A graph is a linear forest if each of its connected components is a path; write $\mathcal{F}_1$ for the class of forests whose maximum degree is at most $1$, and $\mathcal{F}$ for the class of all forests. An $(\mathcal{F}_1,\mathcal{F})$-partition is a partition of the vertex set into two parts inducing, respectively, a graph in $\mathcal{F}_1$ and a graph in $\mathcal{F}$.

Forest-path partition conjecture. Every planar graph without $4$-cycles and $6$-cycles has an $(\mathcal{F}_1,\mathcal{F})$-partition.

This conjecture seeks to strengthen known vertex-partition results for planar graphs excluding 4-cycles and 6-cycles: such graphs are known to admit both a $(\Delta_2,\mathcal{I},\mathcal{I})$-partition and an $(\mathcal{F}_1,\mathcal{I},\mathcal{I})$-partition, while the conjectured partition replaces the two independent-set parts with a single forest and retains a bounded-degree forest part. The source presents this as an open problem.