(1,0,0)-colorability of planar graphs without cycles of length 4 or 6

A graph G is (d1,d2,d3)-colorable if the vertex set V(G) can be partitioned into three subsets V1,V2 and V3 such that for i∈{1,2,3} induced G[Vi] has maximum vertex-degree at most di. So, (0,0,0)-colorability exactly 3-colorability. The well-known Steinberg's conjecture states every planar without cycles of length 4 or 5 3-colorable. As this being disproved by Cohen-Addad etc. in 2017, a similar question, whether i 3-colorable given i∈{6,…,9}, gaining more interest. In paper, we consider question case i=6 from viewpoint improper colorings. More precisely, prove 6 (1,0,0)-colorable, which improves on earlier results they are (2,0,0)-colorable also (1,1,0)-colorable, result graphs to (1,0,0)-colorable.