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.
@article{10_37236_11774,
author = {Ming Han and Xuding Zhu},
title = {Weak (2, 3)-decomposition of planar graphs},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {4},
doi = {10.37236/11774},
zbl = {1533.05213},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11774/}
}
TY - JOUR
AU - Ming Han
AU - Xuding Zhu
TI - Weak (2, 3)-decomposition of planar graphs
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/11774/
DO - 10.37236/11774
ID - 10_37236_11774
ER -
%0 Journal Article
%A Ming Han
%A Xuding Zhu
%T Weak (2, 3)-decomposition of planar graphs
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/11774/
%R 10.37236/11774
%F 10_37236_11774
Ming Han; Xuding Zhu. Weak (2, 3)-decomposition of planar graphs. The electronic journal of combinatorics, Tome 30 (2023) no. 4. doi: 10.37236/11774
