An (F3,F4)-partition of planar graphs without 4- and 6-cycles

Abstract

An $(F_{d_1},\dots ,F_{d_k})$-partition of a graph $G$ is a partition of its vertices set into $k$ subsets $V_1,\dots ,V_k$ where each $V_i$ induces a forest with maximum degree at most $d_i$ for $i\in \{1,\dots ,k\}$. Cho et al.(2021) proved that every planar graph without 4- and 5-cycles admits an $(F_3,F_4)$-partition. In this paper, we show that every planar graph without 4- and 6-cycles admits an $(F_3,F_4)$-partition, which strengthens a previous result due to Nakprasit et al. (2024) in a stronger form.