An (F2,F6)-partition of planar graphs without cycles of length 4 and 6

Let G be a graph with the vertex set $V\left(G\right)$, an $(F_{d_1},\dots ,F_{d_k})$-partition of G is a partition of its vertex set $V\left(G\right)$ into k sets $V_1,\dots ,V_k$ such that the graph $G\left[V_i\right]$ induced by $V_i$ is a forest with maximum degree at most $d_i$ for each $i\in \{1,\dots ,k\}$. Huang et al. (2023) [16] and Sittitrai and Nakprasit, arXiv:2203.06466 [24], independently, showed that every planar graph without cycles of length 4 and 6 has an $(F_2,F)$-partition. Huang et al. (2023) [16] posed a question whether there is a positive integer d such that every planar graph without cycles of length 4 and 6 has an $(F_2,F_d)$-partition. In this paper, we answer affirmatively this question and prove that every planar graph without cycles of length 4 and 6 has an $(F_2,F_6)$-partition, which strengthens the earlier results of Huang, Huang and Lv, and Sittitrai and Nakprasit.