Geodesic
Note on improper coloring of $1$-planar graphs
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 955-968.

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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.
DOI : 10.21136/CMJ.2019.0558-17
Classification : 05C15
Keywords: improper coloring; 1-planar graph; discharging method 
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Chu, Yanan; Sun, Lei; Yue, Jun. Note on improper coloring of $1$-planar graphs. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 955-968. doi : 10.21136/CMJ.2019.0558-17. http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0558-17/

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