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

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 (d1,d2)-coloring of a graph G is a defective two-coloring such that each vertex colored by i has at most di adjacent vertices of the same color, where i=1,2. A graph G is (d1,d2)-colorable if it admits (d1,d2)-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.