Total coloring of planar graphs of maximum degree eight - Roussel, Nicolas; Zhu, Xuding (Information processing letters, 2010)
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 known that if a planar graph G has maximum degree Δ ⩾ 9 , then χ ″ ( G ) = Δ + 1 . Recently Hou et al. (Graphs and Combinatorics 24 (2008) 91–100) proved that if G is a planar graph with maximum degree 8 and with either no 5-cycles or no 6-cycles, then χ ″ ( G ) = 9 . In this Note, we strengthen this result and prove that if G is a planar graph with maximum degree 8, and for each vertex x, there is an integer k x ∈ { 3 , 4 , 5 , 6 , 7 , 8 } such that there is no k x -cycle which contains x, then χ ″ ( G ) = 9 .