Abstract
A total k - coloring of a graph G is a coloring of V ( G ) ∪ E ( G ) using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ ″ ( G ) is the smallest integer k such that G has a total k -coloring. Let G be a planar graph with maximum degree Δ ( G ) and without 6-cycles. In this paper, it is proved that χ ″ ( G ) = Δ ( G ) + 1 if Δ ( G ) ≥ 5 and G contains no 4-cycles, or Δ ( G ) ≥ 6 and G contains no 5-cycles.
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