Total Coloring Conjecture of Planar Graphs Without l-cycles

Given a graph G,a total k-coloring of G is a simultaneous coloring of the vertices and edges of G with at most k colors such that no two incident or adjacent elements receive the same color.Δ(G) is the maximum degree of G.On the total coloring,it has the conjecture that every graph has a total(Δ+2)-coloring.In the paper,it is proved that the total coloring conjecture is true if the planar graph G has no l-cycle and l∈｛3,4,5,6｝.