Planar Graphs without Cycles of Length 3, 4, and 6 are (3, 3)-Colorable

Abstract

For non-negative integers d1 and d2, if V1 and V2 are two partitions of a graph G’s vertex set VG, such that V1 and V2 induce two subgraphs of G, called GV1 with maximum degree at most d1 and GV2 with maximum degree at most d2, respectively, then the graph G is said to be improper d1,d2-colorable, as well as d1,d2-colorable. A class of planar graphs without C3,C4, and C6 is denoted by C. In 2019, Dross and Ochem proved that G is 0,6-colorable, for each graph G in C. Given that d1+d2≥6, this inspires us to investigate whether G is d1,d2-colorable, for each graph G in C. In this paper, we provide a partial solution by showing that G is (3, 3)-colorable, for each graph G in C.