Total colorings of embedded graphs with no 3-cycles adjacent to 4-cycles
Lin Sun, Wang Bing, Jianliang Wu · Discussiones Mathematicae Graph Theory · 2018
A total-k-coloring of a graph G is a coloring of V E using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number (G) of G is the smallest integer k such that G has a total-k-coloring. Let G be a graph embedded in a surface of Euler characteristic 0. If G contains no 3-cycles adjacent to 4-cycles, that is, no 3-cycle has a common edge with a 4-cycle, then (G) max{8, + 1}.