2025/01/13 by Sittitrai, Pongpat, Pimpasalee, Wannapol, Nakprasit, Kittikorn
#05C15 05C10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.07129
A defective k-coloring is a coloring on the vertices of a graph using colors 1,2, …, k such that adjacent vertices may share the same color. A (d1,d2)-coloring of a graph G is a defective 2-coloring of G such that any vertex colored by color i has at most di adjacent vertices of the same color, where i∈\1,2\. A graph G is said to be (d1,d2)-colorable if it admits a (d1,d2)-coloring. Defective 2-coloring in planar graphs without 3-cycles, 4-cycles, and 6-cycles has been investigated by Dross and Ochem, as well as Sittitrai and Pimpasalee. They showed that such graphs are (0,6)-colorable and (3,3)-colorable, respectively. In this paper, we proved that these graphs are also (2,4)-colorable.