Weak (2,3)-Decomposition of Planar Graphs

This paper introduces the concept of weak $(d,h)$-decomposition of a graph $G$, which is defined as a partition of $E(G)$ into two subsets $E_1,E_2$, such that $E_1$ induces a $d$-degenerate graph $H_1$ and $E_2$ induces a subgraph $H_2$ with $\alpha(H_1[N_{H_2}(v)]) \le h$ for any vertex $v$. We prove that each planar graph admits a weak $(2,3)$-decomposition. As a consequence, every planar graph $G$ has a subgraph $H$ such that $G-E(H)$ is $3$-paintable and any proper coloring of $G-E(H)$ is a $3$-defective coloring of $G$. This improves the result in [G. Gutowski, M. Han, T. Krawczyk, and X. Zhu, Defective $3$-paintability of planar graphs, Electron. J. Combin., 25(2):\#P2.34, 2018] that every planar graph is 3-defective $3$-paintable.