Equitable partition of planar graphs

An equitable $k$-partition of a graph $G$ is a collection of induced subgraphs $(G[V_1],G[V_2],\dots ,G[V_k])$ of $G$ such that $(V_1,V_2,\dots ,V_k)$ is a partition of $V(G)$ and $-1\le \mathrm{\mid }{V}_i\mathrm{\mid }-\mathrm{\mid }{V}_j\mathrm{\mid }\le 1$ for all $1\le i<j\le k$. We prove that every planar graph admits an equitable $2$-partition into $3$-degenerate graphs, an equitable $3$-partition into $2$-degenerate graphs, and an equitable $3$-partition into two forests and one graph.