Total Coloring Of Plane Graphs With Maximum Degree At Least 7

The total coloring problem of plane graphs come from famous four coloring prob-lem. Let V, E,Δandδbe the vertex set, edge set, maximum degree and minimum degree of plane graph G, respectively. If one can use k colors to color the elements in V U E such that any two adjacent or incident elements receive distinct colors, then G is said to be k-totally-colorable. At leastΔ+1 colors are needed to color a graph totally. Vizing and Behzad independently conjectured that every graph is (Δ+2)-totally-colorable. This conjecture was known as Total Coloring Conjecture, in short, TCC. But it is unknown whether planar graphs withΔ(G)= 6 is 8-totally-colorable. Going deep into studying in recent years, many scholars in home and abroad found most of planar graphs can reach a lower bound (Δ+1). Borodin, etc. proved that plane graphs withΔ≥11 is (Δ+1)-totally-colorable; Wang Weifan proved that plane graphs withΔ= 10 is (Δ+1)-totally-colorable; Kowalik etc. proved that plane graphs withΔ=9 is (Δ+1)-totally-colorable. Because of K4 is not 4-totally-colorable, Wang Yingqian proposed Total Coloring Conjecture of Plane graphs, in short, PTCC:plane graphs withΔ≥4 is (Δ+1)-totally-colorable. Hou Jianfeng, Shen Lan, etc. proved that some plane graphs withΔ≥8 is (Δ+1)-totally-colorable. Basing on these, I studied mainly on the total coloring problem of planar graphs withΔ≥7 by using a method of discharging and excluding some reducible configurations in this paper.