Note on improper coloring of $1$-planar graphs
Yanan Chu, Lei Sun, Jun Yue · Czechoslovak Mathematical Journal · 2019
summary:A graph $G=(V,E)$ is called improperly $(d_1, \dots , d_k)$-colorable if the vertex set $V$ can be partitioned into subsets $V_1, \dots , V_k$ such that the graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1 \leq i \leq k$. In this paper, we mainly study the improper coloring of $1$-planar graphs and show that $1$-planar graphs with girth at least $7$ are $(2,0,0,0)$-colorable.