Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable

The Total Coloring Conjecture, in short, TCC, says that every simple graph is (Δ+2)(Δ+2)-totally-colorable where ΔΔ is the maximum degree of the graph. Even for planar graphs this conjecture has not been completely settled yet. However, every planar graph with Δ≥9Δ≥9 has been proved to be (Δ+1)(Δ+1)-totally-colorable. In this paper, we prove that planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable.