Relaxed DP-3-coloring of planar graphs without some cycles

Dvo\v{r}\'{a}k and Postle introduced the concept of DP-coloring. DP-coloring is a generalization of list coloring. Sittitrai and Nakprasit combined DP-coloring and defective list coloring to define a new coloring -- relaxed DP-coloring. For relaxed DP-coloring, Nakprasit et al proved that every planar graph without 4- and 7-cycles is DP-(0, 2, 2)-colorable. Li et al proved that every planar graph without 4, 8-cycles or 4, 9-cycles is DP-(1, 1, 1)-colorable. Lu and Zhu proved that every planar graph without 4, 5-cycles, or 4, 6-cycles, or 4, 7-cycles is DP-(1, 1, 1)-colorable. In this paper, we show that every planar graph without 4, 6-cycles or 4, 8-cycles is DP-(0, 2, 2)-colorable.