Relaxed DP-Coloring and another Generalization of DP-Coloring on Planar Graphs without 4-Cycles and 7-Cycles

• Published in: Discussiones Mathematicae. Graph Theory Vol. 43; no. 1; pp. 287 - 297
• Main Authors: Sribunhung, Sarawute, Nakprasit, Keaitsuda Maneeruk, Nakprasit, Kittikorn, Sittitrai, Pongpat
• Format: Journal Article
• Language: English
• Published: Sciendo 01.02.2023; University of Zielona Góra
• Subjects: 05C10; 05C15; discharging; planar graphs; relaxed DP-colorings; variable degeneracy
• ISSN: 1234-3099; 2083-5892
• DOI: 10.7151/dmgt.2405

DP-coloring is generalized via relaxed coloring and variable degeneracy in [P. Sittitrai and K. Nakprasit, , Graphs Combin. 35 (2019) 837–845], [K.M. Nakprasit and K. Nakprasit, , Graphs Combin. 36 (2020) 1189–1201] and [P. Sittitrai and K. Nakprasit, , 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 [ , Appl. Math. Lett. 14 (2001) 269–273] that every planar graph without 4-cycles and 7-cycles is (3, 1)*-choosable.