Geodesic
Relaxed DP-coloring and another generalization of DP-coloring on planar graphs without $4$-cycles and $7$-cycles
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 287-297

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DP-coloring is generalized via relaxed coloring and variable degeneracy in [P. Sittitrai and K. Nakprasit, Sufficient conditions on planar graphs to have a relaxed DP-3-coloring, Graphs Combin. 35 (2019) 837–845], [K.M. Nakprasit and K. Nakprasit, A generalization of some results on list coloring and DP-coloring, Graphs Combin. 36 (2020) 1189–1201] and [P. Sittitrai and K. Nakprasit, An analogue of DP-coloring for variable degeneracy and its applications, Discuss. Math. Graph Theory]. In this work, we introduce another concept that includes two previous generalizations. We demonstrate its application on planar graphs without 4-cycles and 7-cycles. One implication is that the vertex set of every planar graph without 4-cycles and 7-cycles can be partitioned into three sets in which each of them induces a linear forest and one of them is an independent set. Additionally, we show that every planar graph without 4-cycles and 7-cycles is DP-(1,1,1)-colorable. This generalizes a result of Lih et al. [A note on list improper coloring planar graphs, Appl. Math. Lett. 14 (2001) 269–273] that every planar graph without 4-cycles and 7-cycles is (3,1)^∗-choosable.
Keywords: relaxed DP-colorings, variable degeneracy, planar graphs, discharging 
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     title = {Relaxed {DP-coloring} and another generalization of {DP-coloring} on planar graphs without $4$-cycles and $7$-cycles},
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Sribunhung, Sarawute; Nakprasit, Keaitsuda Maneeruk; Nakprasit, Kittikorn; Sittitrai, Pongpat. Relaxed DP-coloring and another generalization of DP-coloring on planar graphs without $4$-cycles and $7$-cycles. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 287-297. http://geodesic.mathdoc.fr/item/DMGT_2023_43_1_a18/