Local condition for planar graphs of maximum degree 7 to be 8-totally colorable
The total chromatic number of a graph G G , denoted by ??(G) ? ? ( G ) , is the minimum number of colors needed to color the vertices and edges of G G such that no two adjacent or incident elements get the same color. It is known that if a planar graph G G has maximum degree ??9 ? ? 9 , then ??(G)=?+1 ? ? ( G ) = ? + 1 . In this paper, we prove that if G G is a planar graph with maximum degree 7 7 , and for every vertex v v , there is an integer kv?{3,4,5,6} k v ? { 3 , 4 , 5 , 6 } so that v v is not incident with any kv k v -cycle, then ??(G)=8 ? ? ( G ) = 8 .
planar graphs total chromatic number total coloring cycle