Weak degeneracy of planar graphs without 4- and 6-cycles

A graph is $k$-degenerate if every subgraph $H$ has a vertex $v$ with $d_{H}(v) \leq k$. The class of degenerate graphs plays an important role in the coloring theory. Observed that $(k + 1)$-choosable and 1)$-DP-colorable. Bernshteyn Lee defined generalization graphs, which called \emph{weakly $k$-degenerate}. weak degeneracy plus one upper bound for many parameters, such as choice number, DP-chromatic number DP-paint number. In this paper, we give two sufficient conditions plane without $4$- $6$-cycles to be weakly $2$-degenerate, implies $3$-DP-colorable near-bipartite, where near-bipartite its set can partitioned into independent acyclic set.