supported by Natural Science Foundation of Shandong Province (Grant No. ZR2009AM009);Scientific Research Foundation for the Excellent Middle-Aged and Youth Scientists of Shandong Province (Grant No. BS2012SF016);National Natural Science Foundation of China (Grant Nos.11001055 and 11101243)
The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum degree Δ≥...