(2, 4)-Colorability of Planar Graphs Excluding Cycles with 3, 4, and 6 Vertices

Pongpat Sittitrai, Wannapol Pimpasalee and Kittikorn Nakprasit ()
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Pongpat Sittitrai: Futuristic Science Research Center, School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand
Wannapol Pimpasalee: Department of Science and Mathematics, Faculty of Science and Health Technology, Kalasin University, Kalasin 46000, Thailand
Kittikorn Nakprasit: Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Mathematics, 2025, vol. 13, issue 11, 1-7

Abstract: A defectivek-coloring is a coloring on the vertices of a graph with colors 1 , 2 , … , k where adjacent vertices may have the same color. A ( d 1 , d 2 ) - coloring of a graph G is a defective two-coloring such that each vertex colored by i has at most d i adjacent vertices of the same color, where i = 1 , 2 . A graph G is ( d 1 , d 2 ) - colorable if it admits ( d 1 , d 2 ) -coloring. For planar graphs excluding cycles with three, four, and six vertices, Dross and Ochem, and additionally Sittitrai and Pimpasalee, have studied their defective 2-coloring. They showed that such graphs are ( 0 , 6 ) - and ( 3 , 3 ) -colorable, respectively. We show in this work that these graphs are also ( 2 , 4 ) -colorable.

Keywords: defective coloring; planar graph; cycle (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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