Geodesic
Total Colorings of Embedded Graphs with No 3-Cycles Adjacent to 4-Cycles
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 4, pp. 977-989

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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 χ′′(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 ε ≥ 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 χ′′(G) ≤ max8, Δ+1.
Keywords: total coloring, embedded graph, cycle 
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Wang, Bing; Wu, Jian-Liang; Sun, Lin. Total Colorings of Embedded Graphs with No 3-Cycles Adjacent to 4-Cycles. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 4, pp. 977-989. http://geodesic.mathdoc.fr/item/DMGT_2018_38_4_a7/